[Volume and circumference]\label{thm:vol-circumference}
Let $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\varphi$. Let $S$ be a closed connected surface in $M$ that is almost transverse to $\varphi$, but is not a fiber. Let $c^{s/u}$ be the stable/unstable multicurves of $S$. Assume that $\mathfrak{c}(M\backslash \!\! \backslash S)<\infty$.

There is a constant $k_S = k_S(\chi(S)) \ge 1$, depending only on $\chi(S)$, and a constant $k_{M \backslash \!\! \backslash S} \ge 1$, depending only on $\chi(S)$ and $\mathfrak c(M\backslash \!\! \backslash S)$, so that the following inequalities hold:
\begin{enumerate}
\item $d_{\mathcal{C}(S)}(c^s, c^u) \le k_S \cdot \mathrm{vol}(M) + k_{M\backslash \!\! \backslash S}$, and
\smallskip
\item $d_{\mathcal{C}(S)}(c^s, c^u) \le k_S \cdot \ell_M(\gamma) + k_{M \backslash \!\! \backslash S}$, where $\gamma$ is any closed geodesic that intersects $S$ essentially and $\ell_M(\gamma)$ is its length in $M$.
\end{enumerate}

\begin{theorem}[Junctures and geometry]\label{cor:finitedepth}
Let $M$ be a closed hyperbolic $3$-manifold with a finite depth foliation $\mc F$ so that $\mc F^0$ is a connected surface $S$. Suppose that $j^\pm  \subset S$ are minimal juncture on $S$.

There is a constant $c_S = c_S(\chi(S)) \ge 1$, depending only on $\chi(S)$, and a constant $D\ge 1$, depending only on $\chi(S)$ and $\mathfrak{c}_{\FF}$, so that the following inequalities hold:
\begin{enumerate}
\item $d_{\C(S)}(j^-, j^+) \le c_S \cdot \vol(M) + D$, and
\smallskip
\item $d_{\C(S)}(j^-, j^+) \le c_S \cdot \ell_M(\gamma) + D$, where $\gamma$ is any closed geodesic that intersects $S$ essentially and $\ell_M(\gamma)$ is its length in $M$.
\end{enumerate}
\end{theorem}

\label{sec:intro}
Let $M$ be a closed hyperbolic $3$--manifold and let $\varphi$ be a pseudo-Anosov flow on $M$. The purpose of this paper is to quantify how dynamical properties of $\varphi$ influence the geometry of $M$ through the curve graph of a closed embedded surface $S\subset M$ that is (almost) transverse to $\varphi$.

\smallskip
The prototypical example is where $S$ is a cross section of $\varphi$ (i.e. $S$ is transverse to $\varphi$ and intersects every orbit), and hence $M$ fibers over the circle with $S$ as a fiber. In this setting, the restriction of the  stable/unstable foliations of $\varphi$ to $S$ are precisely the end invariants of the cover $\widetilde M_S$ of $\widetilde M$ associated to $S$. Hence, by the resolution of Thurston's Ending Lamination Conjecture, proven by Minsky \cite{ECL1} and Brock--Canary--Minsky \cite{ELC2}, the hyperbolic structure on $\widetilde M_S$ is totally determined by these dynamical/geometric invariants. In fact, geometric quantities like the volume of $M$ \cite{Brock2}, the circumference of $M$ \cite{biringer2019ranks, aougab2022covers}, and the location of short geodesics \cite{minsky2001bounded, viaggi2025effective} are each coarsely determined (up to constants depending only on $\chi(S)$) by combinatorial quantities that are organized by the curve graph of $S$ and its subsurfaces.

Despite the significance of these results, very little is known in the more general setting where the surface $S$ is transverse to $\varphi$ but not a cross section (that is, it misses some orbits).
Here, the hyperbolic manifold $\widetilde M_S$ is quasi-Fuchsian (\cite{cooper1994bundles,fenley1999surfaces}) and there is no obvious connection between the stable/unstable foliations of $\varphi$ and the end invariants of $\widetilde M_S$.

\subsection*{Results}

Let $M$ be a closed hyperbolic $3$--manifold, let $\varphi$ be an almost pseudo-Anosov flow on $M$, and let $S$ be a connected surface embedded in $M$ that is transverse to $\varphi$.
 The intersections of $S$ with the invariant stable/unstable foliations $\mathcal F^{s/u}$ of $\varphi$ are a pair of singular foliations $\mathcal F_S^{s/u}$ of $S$.  When $S$ is not a fiber surface (or more precisely, not a cross section of $\varphi$), the foliations $\mathcal F_S^{s/u}$ each contain a nonempty set of closed leaves $c^{s/u}$ (see \Cref{sec:transvese_shadow}).
The pair $c^{s/u}$ are essential multicurves on $S$,
which we call the \emph{stable/unstable multicurves} of $S$.

Informally, our main theorems state that if $c^s$ and $c^u$ are complicated with respect to one another, as viewed in the curve graph of $S$, then the geometry of $M$ near $S$ is also complicated. To state this precisely, we first introduce a notion of complexity of the cut manifold $M\backslash \!\! \backslash S$. This generalizes the notion of `core complexity' originally defined when $S$ is a compact leaf of a depth one foliation of $M$ (see \cite{field2025lower}, \cite{whitfield2024short}, as well as \Cref{sec:examples}).
\smallskip

Recall that $M\backslash \!\! \backslash S$ denotes the manifold obtained by cutting $M$ along $S$. Its boundary is a disjoint union $\partial_- (M \backslash \!\! \backslash S) \sqcup \partial_+ (M \backslash \!\! \backslash S)$, where the flow points into, or out of, $M\backslash \!\! \backslash S$ respectively. A \emph{product annulus} in $M \backslash \!\! \backslash S$ is an essential properly embedded annulus that has one boundary component on each of $\partial_-(M\backslash \!\! \backslash S)$ and $\partial_+(M\backslash \!\! \backslash S)$.
Define the \emph{core complexity} of $M\backslash \!\! \backslash S$ to be
\begin{align*}
\mathfrak{c}(M\backslash \!\! \backslash S) =
\begin{cases}
0, & M\backslash \!\! \backslash S \text{ contains a product annulus},\\
\min{-\chi(\Sigma)}, & \text{otherwise},
\end{cases}
\end{align*}
where the min is over all properly embedded surfaces $\Sigma$ in $M\backslash \!\! \backslash S$ that are transverse to (a dynamic blowup of) $\varphi$ and which meet both $\partial_+ (M \backslash \!\! \backslash S)$ and $\partial_- (M\backslash \!\! \backslash S)$. Dynamic blowups and almost pseudo-Anosov flows are defined in \Cref{sec:pA}.
For context, $\mathfrak{c}(M\backslash \!\! \backslash S)$ contributes to an \emph{additive} error term in our theorems that relate geometry and dynamics. We note that $\mathfrak{c}(M\backslash \!\! \backslash S)< \infty$ when $\varphi$ is a suspension flow (i.e. $\varphi$ admits a cross section) or more generally when $S$ is a leaf of a (almost) transverse finite depth foliation (see \Cref{sec:examples}).
A cohomological characterization of when $\mathfrak{c}(M\backslash \!\! \backslash S)< \infty$ will be  subject of forthcoming work \cite{HT_depthone}.

\smallskip

Our first main theorem relates the curve graph distance between $c^u$ and $c^s$ to the volume and circumference of $M$. See \Cref{th:vol} and \Cref{th:circum}, respectively.

\label{sec:intro}
Let $M$ be a closed hyperbolic $3$--manifold and let $\varphi$ be a pseudo-Anosov flow on $M$. The purpose of this paper is to quantify how dynamical properties of $\varphi$ influence the geometry of $M$ through the curve graph of a closed embedded surface $S\subset M$ that is (almost) transverse to $\varphi$.

\smallskip
The prototypical example is where $S$ is a cross section of $\varphi$ (i.e. $S$ is transverse to $\varphi$ and intersects every orbit), and hence $M$ fibers over the circle with $S$ as a fiber. In this setting, the restriction of the stable/unstable foliations of $\varphi$ to $S$ are precisely the end invariants of the cover $\widetilde M_S$ of $\widetilde M$ associated to $S$. Hence, by the resolution of Thurston's Ending Lamination Conjecture, proven by Minsky \cite{ECL1} and Brock--Canary--Minsky \cite{ELC2}, the hyperbolic structure on $\widetilde M_S$ is totally determined by these dynamical/geometric invariants. In fact, geometric quantities like the volume of $M$ \cite{Brock2}, the circumference of $M$ \cite{biringer2019ranks, aougab2022covers}, and the location of short geodesics \cite{minsky2001bounded, viaggi2025effective} are each coarsely determined (up to constants depending only on $\chi(S)$) by combinatorial quantities that are organized by the curve graph of $S$ and its subsurfaces.

Let $M$ be a closed hyperbolic $3$--manifold, let $\varphi$ be an almost pseudo-Anosov flow on $M$, and let $S$ be a connected surface embedded in $M$ that is transverse to $\varphi$.
 The intersections of $S$ with the invariant stable/unstable foliations $\mathcal F^{s/u}$ of $\varphi$ are a pair of singular foliations $\mathcal F_S^{s/u}$ of $S$. When $S$ is not a fiber surface (or more precisely, not a cross section of $\varphi$), the foliations $\mathcal F_S^{s/u}$ each contain a nonempty set of closed leaves $c^{s/u}$ (see \Cref{sec:transvese_shadow}).
The pair $c^{s/u}$ are essential multicurves on $S$,
which we call the \emph{stable/unstable multicurves} of $S$.

Recall that $M\backslash \!\! \backslash S$ denotes the manifold obtained by cutting $M$ along $S$. Its boundary is a disjoint union $\partial_- (M \backslash \!\! \backslash S) \sqcup \partial_+ (M \backslash \!\! \backslash S)$, where the flow points into, or out of, $M\backslash \!\! \backslash S$ respectively. A \emph{product annulus} in $M \backslash \!\! \backslash S$ is an essential properly embedded annulus that has one boundary component on each of $\partial_-(M\backslash \!\! \backslash S)$ and $\partial_+(M\backslash \!\! \backslash S)$.
Define the \emph{core complexity} of $M\backslash \!\! \backslash S$ to be
\begin{align*}
\mathfrak{c}(M\backslash \!\! \backslash S) =
\begin{cases}
0, & M\backslash \!\! \backslash S \text{ contains a product annulus},\\
\min{-\chi(\Sigma)}, & \text{otherwise},
\end{cases}
\end{align*}
where the min is over all properly embedded surfaces $\Sigma$ in $M\backslash \!\! \backslash S$ that are transverse to (a dynamic blowup of) $\varphi$ and which meet both $\partial_+ (M \backslash \!\! \backslash S)$ and $\partial_- (M\backslash \!\! \backslash S)$. Dynamic blowups and almost pseudo-Anosov flows are defined in \Cref{sec:pA}.
For context, $\mathfrak{c}(M\backslash \!\! \backslash S)$ contributes to an \emph{additive} error term in our theorems that relate geometry and dynamics. We note that $\mathfrak{c}(M\backslash \!\! \backslash S)< \infty$ when $\varphi$ is a suspension flow (i.e. $\varphi$ admits a cross section) or more generally when $S$ is a leaf of a (almost) transverse finite depth foliation (see \Cref{sec:examples}).
A cohomological characterization of when $\mathfrak{c}(M\backslash \!\! \backslash S)< \infty$ will be subject of forthcoming work \cite{HT_depthone}.

\smallskip

Our first main theorem relates the curve graph distance between $c^u$ and $c^s$ to the volume and circumference of $M$. See \Cref{th:vol} and \Cref{th:circum}, respectively.

\begin{theorem}[Circumference]
\label{th:circum}
Let $M$ be a closed hyperbolic $3$-manifold, $\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\varphi$ but not a fiber. Then there is a constant $D \ge 0$, depending only on $|\chi(S)|$ and $\mf c(M\cut S)$, so that
\[
d_{\C(S)}(c^s, c^u) \prec  \ell(\gamma) + D,
\]
 where $\gamma$ is any closed geodesic that intersects $S$ essentially.
\end{theorem}

\begin{theorem}[Volume]\label{th:vol}
Let $M$ be a closed hyperbolic $3$-manifold, $\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\varphi$ but not a fiber.
Then
\[
d_{\C(S)}(c^s, c^u) \prec \vol(M) + D_{M\cut S}
\]
\end{theorem}

\smallskip
The prototypical example is where $S$ is a cross section of $\varphi$ (i.e. $S$ is transverse to $\varphi$ and intersects every orbit), and hence $M$ fibers over the circle with $S$ as a fiber. In this setting, the restriction of the stable/unstable foliations of $\varphi$ to $S$ are precisely the end invariants of the cover $\wt M_S$ of $\wt M$ associated to $S$. Hence, by the resolution of Thurston's Ending Lamination Conjecture, proven by Minsky \cite{ECL1} and Brock--Canary--Minsky \cite{ELC2}, the hyperbolic structure on $\wt M_S$ is totally determined by these dynamical/geometric invariants. In fact, geometric quantities like the volume of $M$ \cite{Brock2}, the circumference of $M$ \cite{biringer2019ranks, aougab2022covers}, and the location of short geodesics \cite{minsky2001bounded, viaggi2025effective} are each coarsely determined (up to constants depending only on $\chi(S)$) by combinatorial quantities that are organized by the curve graph of $S$ and its subsurfaces.

Our first main theorem relates the curve graph distance between $c^u$ and $c^s$ to the volume and circumference of $M$. See \Cref{th:vol} and \Cref{th:circum}, respectively.

We remark that when $M\cut S$ contains a product annulus, $\mf c(M\cut S) = 0$ and so the constants in \Cref{thm:vol-circumference} depend only on $\vert \chi(S) \vert$.

\begin{thm}[Short curves]\label{thm:short-curves}
Let $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\varphi$. Let $S$ be a closed surface in $M$ that is almost transverse to $\varphi$ and let $Y \subset S$ be a
subsurface of $S$. Assume that $\mathfrak{c}(M\cut S)<\infty$. Then for any $\epsilon >0$, there is a $K = K(\epsilon, \chi(S)) \ge 0$, depending only on $\epsilon$ and $\chi(S)$, such that if
\begin{itemize}
\item $d_{\C(S)}(c^{s/u} ,\partial Y) \ge k_{M\cut S}$ and
\item $d_{\C(Y)}(c^s,c^u) \ge K + 2 \cdot k_{M \cut S}$,
\end{itemize}
then
\[
\ell_M(\partial Y) \le \epsilon.
\]
\end{thm}

\begin{theorem}[Volume]\label{th:vol}
Let $M$ be a closed hyperbolic $3$-manifold, $\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\varphi$ but not a fiber.
Then
\[
d_{\C(S)}(c^s, c^u) \prec \vol(M) + D_{M\cut S}
\]
\end{theorem}

\begin{theorem}[Circumference]
\label{th:circum}
Let $M$ be a closed hyperbolic $3$-manifold, $\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\varphi$ but not a fiber. Then there is a constant $D \ge 0$, depending only on $|\chi(S)|$ and $\mf c(M\cut S)$, so that
\[
d_{\C(S)}(c^s, c^u) \prec \ell(\gamma) + D,
\]
 where $\gamma$ is any closed geodesic that intersects $S$ essentially.
\end{theorem}

\subsection{Short curves}
Finally, we give a short proof of the following:
\begin{theorem}[Short curves] \label{th:short_curves}
Let $M$ be a closed hyperbolic $3$-manifold with an almost pseudo-Anosov flow $\varphi$. Let $S$ be a closed surface in $M$ that is transverse to $\varphi$ and let $Y \subset S$ be a
subsurface of $S$. Then for any $\epsilon >0$, there is a $K = K(\epsilon, \chi(S)) \ge 0$, depending only on $\epsilon$ and $\chi(S)$, such that if
\begin{itemize}
\item $d_{\C(S)}(c^{s/u} ,\partial Y) \ge 2D_{M\cut S}+2$ and
\item $d_{\C(Y)}(c^s,c^u) \ge K + 4 \cdot D_{M \cut S}$,
\end{itemize}
then
\[
\ell_M(\partial Y) \le \epsilon.
\]
\end{theorem}

There is a constant $c_S = c_S(\chi(S)) \ge 1$, depending only on $\chi(S)$, and a constant $D\ge 1$, depending only on $\chi(S)$ and $\mathfrak{c}_{\FF}$, so that the following inequalities hold:
\begin{enumerate}
\item $d_{\C(S)}(j^-, j^+) \le c_S \cdot \vol(M) + D$, and
\smallskip
\item $d_{\C(S)}(j^-, j^+) \le c_S \cdot \ell_M(\gamma) + D$, where $\gamma$ is any closed geodesic that intersects $S$ essentially and $\ell_M(\gamma)$ is its length in $M$.
\end{enumerate}
\end{theorem}

\begin{theorem}[Circumference]
\label{th:circum}
Let $M$ be a closed hyperbolic $3$-manifold, $\varphi$ an almost pseudo-Anosov flow on $M$, and $S$ a closed surface that is transverse to $\varphi$ but not a fiber. Then there is a constant $D \ge 0$, depending only on $|\chi(S)|$ and $\mf c(M\cut S)$, so that
\[
d_{\C(S)}(c^s, c^u) \prec  \ell(\gamma) + D,
\]
 where $\gamma$ is any closed geodesic that intersects $S$ essentially.
\end{theorem}

\begin{thm}[Volume and circumference]\label{thm:vol-circumference}
Let $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\varphi$. Let $S$ be a closed connected surface in $M$ that is almost transverse to $\varphi$, but is not a fiber. Let $c^{s/u}$ be the stable/unstable multicurves of $S$. Assume that $\mathfrak{c}(M\cut S)<\infty$.

There is a constant $k_S = k_S(\chi(S)) \ge 1$, depending only on $\chi(S)$, and a constant $k_{M \cut S} \ge 1$, depending only on $\chi(S)$ and $\mf c(M\cut S)$, so that the following inequalities hold:
\begin{enumerate}
\item $d_{\C(S)}(c^s, c^u) \le k_S \cdot \vol(M) + k_{M\cut S}$, and
\smallskip
\item $d_{\C(S)}(c^s, c^u) \le k_S \cdot \ell_M(\gamma) + k_{M \cut S}$, where $\gamma$ is any closed geodesic that intersects $S$ essentially and $\ell_M(\gamma)$ is its length in $M$.
\end{enumerate}
\end{thm}

We remark that when $M\backslash \!\! \backslash S$ contains a product annulus, $\mathfrak c(M\backslash \!\! \backslash S) = 0$ and so the constants in \Cref{thm:vol-circumference} depend only on $\vert \chi(S) \vert$.

There is also a version of \Cref{thm:vol-circumference} that starts with a finite depth foliation rather than a flow, where the multicurves $c^{s/u}$ are replaced by minimal junctures of the foliation. See \Cref{sec:examples} and in particular \Cref{cor:finitedepth} for details.

\smallskip
The second main theorem relates subsurface distances between $c^u$ and $c^s$ to short geodesics in $M$. Informally, it says that if $c^{s/u}$ are complicated with respect to one another, from the perspective of some distant subsurface $Y \subset S$, then the geodesic length of $\partial Y$ in $M$ must be small. See \Cref{th:short_curves}.

\begin{thm}[Short curves]\label{thm:short-curves}
Let $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\varphi$. Let $S$ be a closed surface in $M$ that is almost transverse to $\varphi$ and let $Y \subset S$ be a
subsurface of $S$. Assume that $\mathfrak{c}(M\backslash \!\! \backslash S)<\infty$. Then for any $\varepsilon >0$, there is a $K = K(\varepsilon, \chi(S)) \ge 0$, depending only on $\varepsilon$ and $\chi(S)$, such that if
\begin{itemize}
\item $d_{\mathcal{C}(S)}(c^{s/u} ,\partial Y) \ge k_{M\backslash \!\! \backslash S}$ and
\item $d_{\mathcal{C}(Y)}(c^s,c^u) \ge K + 2 \cdot  k_{M \backslash \!\! \backslash S}$,
\end{itemize}
then
\[
\ell_M(\partial Y) \le \varepsilon.
\]
\end{thm}

Here, $\ell_M(\partial Y)$ denotes the length of the geodesic representative of $\partial Y$ in $M$, and $d_{\mathcal{C}(Y)}(c^s,c^u)$ is the subsurface distance in the curve graph of $Y$ between $c^s$ and $c^u$. See \Cref{sec:curvegraph} for details.

\begin{remark}
Our main technical result that controls the geometry of $M$ is \Cref{prop:bounded_length}. Informally, it states that each of the multicurves $c^{s/u}$ have bounded curve graph distance from curves on $S$ that have bounded hyperbolic length in $M$. One might hope to prove the stronger statement that $c^{s/u}$ themselves have bounded length, but this is not always possible. Indeed, Whitfield \cite[Theorem A]{whitfield2024short} produces examples where the `junctures' of a depth one foliation of $M$ are arbitrarily short. (Here, the junctures are essentially boundary components of a properly embedded surface in $M \backslash \!\! \backslash S$ realizing the minimum core complexity.)
Since such curves necessarily cross $c^{s/u}$, in these examples the stable/unstable multicurves can be arbitrarily long.
\end{remark}

Finally, several examples are given in \Cref{sec:examples}, including a flexible construction of manifolds and flows that realizes the hypotheses of the main theorems.

\subsection{Connections to the literature}
Although most of the previous work that relates hyperbolic geometry, dynamics, and the curve graph comes from the fibered manifold setting (as discussed above), there has recently been an interest in extending these connections to mapping tori of endperiodic maps of infinite-type surfaces. This is most prominent in work of Field, Kim, Kent, Leininger, and Loving \cite{field2023end, field2025lower} who establish connections to hyperbolic volume of the mapping tori, and work of Whitfield \cite{whitfield2024short}, who studies the lengths of short geodesics.

The work in this paper sheds light on a different aspect of the geometry of these mapping tori. Indeed, returning to the setting of our main theorems, in the special case where $S$ is a compact leaf of a transverse depth one foliation of $M$, each component of $M \backslash \!\! \backslash S$ is the mapping torus of an especially nice endperiodic map -- the first return map to a depth one leaf of the foliation using the transverse flow.
The works cited above study the intrinsic geometry of components of $M \backslash \!\! \backslash S$, especially for the unique totally geodesic structure, when it exists. By contrast, our results here are about the geometry of the original closed manifold $M$. Informally,  \Cref{thm:vol-circumference} and \Cref{thm:short-curves} focus on the geometric features of $M$ that are forced when gluing $M \backslash \!\! \backslash S$ back to obtain $M$, as measured by the curve graph complexity of the stable and unstable multicurves. For more on the implications in the depth one case, see \Cref{sec:examples}.

\subsection{Proof strategy and outline of paper}
The proof of \Cref{thm:vol-circumference} and \Cref{thm:short-curves} is divided into three steps. In the first step, we identify multicurves on $S$ that have topological significance in $M\backslash \!\! \backslash S$, and have bounded intersection number with $c^{s/u}$. When $M\backslash \!\! \backslash S$ admits product annuli, these are the components of Thurston's window frame of $M\backslash \!\! \backslash S$ (see \Cref{sec:geom}),
and we show that product annuli can be put into a particularly nice position with respect to the flow. This is explained in \Cref{sec:annuli}.
When $M\backslash \!\! \backslash S$ does not admit a product annulus but has finite core complexity, these are the boundary components of a properly embedded transverse surface with minimal complexity. In the first case, the intersection number bounds are established in \Cref{sec:annuli}, and in the second case, this is done in \Cref{sec:int_bounds}.

In the second step, we show that these multicurves have bounded length in the hyperbolic metric on $M$. This is done in \Cref{sec:geom}.
The results in the first two steps lead to \Cref{prop:bounded_length}, which states that $c^s$ and $c^u$ have bounded intersection number with bounded length curves in $S$. Finally, we obtain geometric control of $M$ by passing to the quasi-Fuchsian cover corresponding to $S$ and using standard tools from the study of curve graphs and Kleinian surface groups, replacing $c^s$ and $c^u$ with nearby bounded length curves. This is done in \Cref{sec:endgame}.

Finally,
examples and applications are given in \Cref{sec:examples}, including a version of \Cref{thm:vol-circumference} for finite depth foliations as well as a construction where the
dynamical quantity $d_{\mathcal{C}(S)}(c^s,c^u)$ can be precisely controlled.

\subsection*{Acknowledgements}
The authors thank Michael Landry and Yair Minsky for several helpful and encouraging conversations. The authors also thank Ellis Buckminster and Brandis Whitfield for helpful comments on an earlier draft.

The completion of this paper was supported by the National Science Foundation under Grant No. DMS--2424139, while the authors were in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the Spring 2026 semester.
Taylor was also partially supported by NSF grant DMS--2503113 and the Simons Foundation, and Huang was partially supported by NSF grant DMS-2005328.

The proof of \Cref{thm:vol-circumference} (and \Cref{thm:short-curves}) is described as follows: it is "divided into three steps." (1) "In the first step, we identify multicurves on $S$ that have topological significance in $M\backslash \!\! \backslash S$, and have bounded intersection number with $c^{s/u}$." If $M\backslash \!\! \backslash S$ admits product annuli, these multicurves are "the components of Thurston's window frame" and the authors "show that product annuli can be put into a particularly nice position with respect to the flow" (\Cref{sec:annuli}); if there are no product annuli but finite core complexity, they instead use "the boundary components of a properly embedded transverse surface with minimal complexity" and establish the intersection bounds in \Cref{sec:int_bounds}. (2) "In the second step, we show that these multicurves have bounded length in the hyperbolic metric on $M$" (\Cref{sec:geom}). Steps (1) and (2) yield \Cref{prop:bounded_length}, stating that "$c^s$ and $c^u$ have bounded intersection number with bounded length curves in $S$." (3) "Finally, we obtain geometric control of $M$ by passing to the quasi-Fuchsian cover corresponding to $S$ and using standard tools from the study of curve graphs and Kleinian surface groups, replacing $c^s$ and $c^u$ with nearby bounded length curves" (\Cref{sec:endgame}).

The proof of the main theorem (and the following theorem) is described as follows: it is "divided into three steps."

We first state the following theorem.
\begin{thm}[Short curves]\label{thm:short-curves}
Let $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\varphi$. Let $S$ be a closed surface in $M$ that is almost transverse to $\varphi$ and let $Y \subset S$ be a
subsurface of $S$. Assume that $\mathfrak{c}(M\cut S)<\infty$. Then for any $\epsilon >0$, there is a $K = K(\epsilon, \chi(S)) \ge 0$, depending only on $\epsilon$ and $\chi(S)$, such that if
\begin{itemize}
\item $d_{\C(S)}(c^{s/u} ,\partial Y) \ge k_{M\cut S}$ and
\item $d_{\C(Y)}(c^s,c^u) \ge K + 2 \cdot  k_{M \cut S}$,
\end{itemize}
then
\[
\ell_M(\partial Y) \le \epsilon.
\]
\end{thm}

(1) "In the first step, we identify multicurves on $S$ that have topological significance in $M\backslash \!\! \backslash S$, and have bounded intersection number with $c^{s/u}$." If $M\backslash \!\! \backslash S$ admits product annuli, these multicurves are "the components of Thurston's window frame" and the authors "show that product annuli can be put into a particularly nice position with respect to the flow" (\Cref{sec:annuli}); if there are no product annuli but finite core complexity, they instead use "the boundary components of a properly embedded transverse surface with minimal complexity" and establish the intersection bounds in \Cref{sec:int_bounds}.

(2) "In the second step, we show that these multicurves have bounded length in the hyperbolic metric on $M$" (\Cref{sec:geom}). Steps (1) and (2) yield the following proposition, stating that "$c^s$ and $c^u$ have bounded intersection number with bounded length curves in $S$."
\begin{proposition}\label{prop:bounded_length}
There are constants $L, d \ge 1$, depending on $|\chi(S)|$ and $\mf c(M \cut S)$, and curves $\alpha$ and $\beta$ in $\C(S)$ such that:
\begin{itemize}
\item the lengths of the geodesic representatives of $\alpha$ and $\beta$ in $M$ are no more than $L$, and
\item the intersection numbers $i(c^s, \alpha)$ and $i(c^u, \beta)$ are no more than $d$.
\end{itemize}
\end{proposition}

(3) "Finally, we obtain geometric control of $M$ by passing to the quasi-Fuchsian cover corresponding to $S$ and using standard tools from the study of curve graphs and Kleinian surface groups, replacing $c^s$ and $c^u$ with nearby bounded length curves" (\Cref{sec:endgame}).

Existential–Universal
Inequality or Bound

Question: Let $M$ be a closed hyperbolic $3$-manifold with a pseudo-Anosov flow $\varphi$, and let $S\subset M$ be a closed connected surface that is almost transverse to $\varphi$ but is not a fiber surface. Let $c^s$ and $c^u$ denote the stable and unstable multicurves of $S$, meaning the essential multicurves formed by the closed leaves in the singular foliations induced on $S$ by the stable and unstable foliations of $\varphi$. Write $\mathcal C(S)$ for the curve graph of $S$ and $d_{\mathcal C(S)}$ for its distance. Let $M\backslash\!\!\backslash S$ be the manifold obtained by cutting $M$ along $S$, and assume its core complexity $\mathfrak c(M\backslash\!\!\backslash S)$ is finite (where $\mathfrak c=0$ if $M\backslash\!\!\backslash S$ contains a product annulus, and otherwise $\mathfrak c$ is the minimum of $-\chi(\Sigma)$ over properly embedded surfaces $\Sigma$ that are transverse to a dynamic blowup of $\varphi$ and meet both boundary sides of $M\backslash\!\!\backslash S$). Which explicit uniform estimate is guaranteed?

There exist constants $k_S=k_S(\chi(S))\ge 1$, depending only on $\chi(S)$, and $k_{M\backslash\!\!\backslash S}\ge 1$, depending only on $\chi(S)$ and $\mathfrak c(M\backslash\!\!\backslash S)$, such that both
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\mathrm{vol}(M)+k_{M\backslash\!\!\backslash S}
\]
and, for every closed geodesic $\gamma$ in $M$ that intersects $S$ essentially,
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\ell_M(\gamma)+k_{M\backslash\!\!\backslash S},
\]
where $\ell_M(\gamma)$ is the hyperbolic length of $\gamma$ in $M$.

Choices:
B.
There exist constants $k_S=k_S(\chi(S))\ge 1$, depending only on $\chi(S)$, and $k_{M\backslash\!\!\backslash S}\ge 1$, depending only on $\chi(S)$ and $\mathfrak c(M\backslash\!\!\backslash S)$, such that both
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\mathrm{vol}(M)+k_{M\backslash\!\!\backslash S}
\]
and, for every closed geodesic $\gamma$ in $M$ that intersects $S$ essentially,
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\ell_M(\gamma),
\]
where $\ell_M(\gamma)$ is the hyperbolic length of $\gamma$ in $M$.
C.
There exist constants $k_S=k_S(\chi(S))\ge 1$ and $k_{M\backslash\!\!\backslash S}\ge 1$, depending only on $\chi(S)$ and $\mathfrak c(M\backslash\!\!\backslash S)$, such that
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\mathrm{vol}(M)+k_{M\backslash\!\!\backslash S}.
\]
D.
There exist constants $k_S=k_S(\chi(S))\ge 1$ and $k_{M\backslash\!\!\backslash S}\ge 1$, depending only on $\chi(S)$ and $\mathfrak c(M\backslash\!\!\backslash S)$, such that both
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\mathrm{vol}(M)+k_{M\backslash\!\!\backslash S}
\]
and, for some closed geodesic $\gamma$ in $M$ that intersects $S$ essentially,
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\ell_M(\gamma)+k_{M\backslash\!\!\backslash S},
\]
where $\ell_M(\gamma)$ is the hyperbolic length of $\gamma$ in $M$.
E.
There exist constants $k_S=k_S(\chi(S))\ge 1$, depending only on $\chi(S)$, and $k_{M\backslash\!\!\backslash S}\ge 1$, depending only on $\chi(S)$ and $\mathfrak c(M\backslash\!\!\backslash S)$, such that both
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\mathrm{vol}(M)+k_{M\backslash\!\!\backslash S}
\]
and, for every closed geodesic $\gamma$ in $M$ that intersects $S$ essentially,
\[
d_{\mathcal C(S)}(c^s,c^u)\le k_S\,\ell_M(\gamma)+k_{M\backslash\!\!\backslash S},
\]
with $k_{M\backslash\!\!\backslash S}$ depending only on $\mathfrak c(M\backslash\!\!\backslash S)$ and not on $\chi(S)$.

additive error from bounded-length replacement curves

dropped the second inequality involving arbitrary essential geodesics

universality over geodesics intersecting S essentially

dependence of the additive constant on both chi(S) and core complexity


\label{Thm_Uniform_bound}
		Let $\Omega=B_1(0)$ be the unit ball in $\mathbb R^2$, assume conditions $(H_1)$, $(H_2)$, $(H_3)$, $(H_b)$, and alternatively
		\begin{enumerate}
			\item[i)] \eqref{BM} holds;
			\item[ii)] \eqref{nonBM}, $(H_4)$, $(H_5)$, and either $(H_6)$ or $(H_6')$ hold.
		\end{enumerate}
		Then there exists a constant $C>0$ such that
		\begin{equation*}
			\|u\|_\infty\leq C\qquad\mbox{and}\qquad\|v\|_\infty\leq C
		\end{equation*}
		for all (eventual) solutions $(u,v)$ of system \eqref{sys}.

\begin{equation}\label{NL_model}
		f(t)\sim\etal,\quad g(t)\sim\etb\qquad\mbox{with}\ \ \alpha>0,\ \beta>0
	\end{equation}

\begin{thm}\label{Thm_Uniform_bound}
		Let $\Omega=B_1(0)$ be the unit ball in $\R^2$, assume conditions $(H_1)$, $(H_2)$, $(H_3)$, $(H_b)$, and alternatively
		\begin{enumerate}
			\item[i)] \eqref{BM} holds;
			\item[ii)] \eqref{nonBM}, $(H_4)$, $(H_5)$, and either $(H_6)$ or $(H_6')$ hold.
		\end{enumerate}
		Then there exists a constant $C>0$ such that
		\begin{equation*}
			\|u\|_\infty\leq C\qquad\mbox{and}\qquad\|v\|_\infty\leq C
		\end{equation*}
		for all (eventual) solutions $(u,v)$ of system \eqref{sys}.
	\end{thm}

\begin{lem}\label{growth_below_g}
		Under \eqref{BM} or \eqref{nonBM}, there exist $C_1,C_2>0$ and $\delta_1,\delta_2>0$ such that $f(s)<C_1\e^{\delta_1s}$ and $g(s)>C_2\e^{\delta_2s}$ for $s$ large enough.
	\end{lem}

\begin{lem}\label{TM-subcritiche}
		Under \eqref{BM} or \eqref{nonBM}, condition $(H_3)$ implies that for all $p>0$ one has
        $$\lim_{t\to+\infty}\frac{f(t)}{t^p}=+\infty=\lim_{t\to+\infty}\frac{g(t)}{t^p}$$
        and, for all $\gamma>0$,
        $$\lim_{t\to+\infty}\frac{f(t)}{\e^{\gamma t^2}}=0=\lim_{t\to+\infty}\frac{g(t)}{\e^{\gamma t^2}}\,.$$
	\end{lem}

\begin{lem}\label{lem_f8_conseq}
		Suppose \eqref{nonBM}, then
		\begin{enumerate}
			\item[i)] if $(H_6)$ is assumed, then $f(t)\ges\e^{\frac a3(\log t)^3}$ for $t$ large;
			\item[ii)] if $(H_6')$ is assumed, then $f(t)\ges\e^{\frac a2\log t(\loglog t)^2}\,$ for $t$ large.
		\end{enumerate}
	\end{lem}

Let $\Omega\subset\mathbb R^N$ be a smooth bounded domain and consider the semilinear Hamiltonian elliptic system with Dirichlet boundary conditions
    \begin{equation}\label{sys}
		\begin{cases}
			-\Delta u=f(v)\quad&\mbox{in }\Omega\,,\\
			-\Delta v=g(u)\quad&\mbox{in }\Omega\,,\\
			u=v=0\quad&\mbox{on }{\partial\Omega}\,.
		\end{cases}
	\end{equation}
    It is well known that for $N\geq3$ polynomial growth conditions at infinity have to be fulfilled in order to set these variational problems in Sobolev spaces and prove existence.
    If $N\geq3$, for systems of the kind \eqref{sys}, the good notion of criticality is represented by a critical hyperbola: in the model case $f(v)=|v|^{p-1}v$ and $g(u)=|u|^{q-1}u$ with $p,q>1$, then the critical hyperbola defined as
    \begin{equation}\label{hyp}
        \frac1{p+1}+\frac1{q+1}=\frac{N-2}N
    \end{equation}
    divides existence, above \eqref{hyp}, \cite{HV,HMV}, from nonexistence, below \eqref{hyp}, \cite{M}, of classical positive solutions. Note that if \eqref{sys} is set in the entire $\mathbb R^N$ space, then it is proved that the Sobolev hyperbola \eqref{hyp} plays the same role for $N=3,4$, while it is still an open problem, under the name of Lane-Emden conjecture, for dimensions $N\ge 5$, see \cite{M96,PQS,S}. In contrast, much faster growths are admissible in dimension two: in particular, exponential nonlinearities can be treated, with the Trudinger-Moser inequality taking the role of the Sobolev embedding theorem valid for $N\geq3$. For the system \eqref{sys} we still can find a notion of critical hyperbola in the setting of Sobolev-Lorentz spaces: to give an idea, if $f(t)\sim{\rm e}^{|t|^\alpha}$ and $g(t)\sim{\rm e}^{|t|^\beta}$ with $0<\alpha,\beta<+\infty$ (here the symbol $\sim$ means having the same asymptotic growth), then the maximal growth is obtained on the ``conformal'' critical hyperbola
    \begin{equation}\label{abhyp}
        \frac1\alpha+\frac1\beta=1\,,
    \end{equation}
    see \cite{Ruf_sys} and the more recent advance in \cite{dORR}.

    When one treats equations or systems which are not necessarily variational (e.g. \eqref{sys} when $f$ and $g$ depend on both variables $u$ and $v$), a typical technique to prove the existence of solutions is the topological degree theory. An essential step in the proof is to provide a priori uniform bounds for solutions in $L^\infty$ norm. In fact, from this one infers that a certain fixed point (or Leray-Schauder) index in a large ball of the functional space is zero; moreover, by imposing some natural assumptions on the nonlinearities at $0$, the index in a neighbourhood of the origin is not zero, and therefore, by additivity of the index, one finds a nontrivial solution.

    This strategy has been extensively employed in various contexts, and the study of $L^\infty$ a priori bounds for solutions has become a topic of independent interest in the literature. For the scalar case, we refer to the classical papers by Gidas and Spruck \cite{GS} and de Figueiredo, Lions, and Nussbaum \cite{dFLN} for semilinear problems involving subcritical polynomial nonlinearities in dimension $N\geq3$. Such results have been extended to the higher-order and quasilinear settings, and for wider classes of subcritical nonlinearities, see e.g. \cite{RW,AC,Ruiz,Zou,BF_eq,DP}.	In broad terms, these results indicate that a priori bounds for strong as well as distributional solutions can be established for (almost) all subcritical nonlinearities, up to the threshold of Sobolev critical growth. Concerning the system \eqref{sys} with $N\geq3$, the situation is comparable, in the sense that the critical hyperbola still represents the threshold for both existence and $L^\infty$ a priori bounds, see among others \cite{CdFM,Zou_sys,QS,PQS,BF}.

	The situation considerably changes in the case of the \textit{conformal dimension}, where the critical threshold for the existence $t\mapsto{\rm e}^{t^{\frac N{N-1}}}$ is given by the Trudinger-Moser inequality. Here, as the seminal work of Brezis-Merle \cite{BM} shows for the scalar second-order case in dimension $N=2$, the limiting growth up to which one may expect to find a priori bounds within the class of \textit{distributional} solutions of $-\Delta u=f(x,u)$ with Dirichlet boundary conditions is only $t\mapsto{\rm e}^t$. Within this setting, in \cite{BM} the authors prove a priori bounds for nonlinearities bounded from below and above by ${\rm e}^t$, under the assumption $\int_\Omega f(x,u)\, {\rm d} x\leq\Lambda$; see also \cite{ChenLi}. We note that such a condition follows from the analysis near the boundary already obtained in \cite{dFLN}. More in general, the respective quasilinear problem involving $-\Delta_N$ in $\Omega\subset\mathbb R^N$ with $N\geq2$, was investigated in \cite{LRU}, where the nonlinearities behave either like ${\rm e}^{pt}$ for some $p>0$, or grow less than ${\rm e}^{t^\alpha}$ with $\alpha\in(0,1)$. The authors use regularity estimates in Orlicz spaces to cover the latter case, while extend the arguments from \cite{BM} in the former. This diversity of the method causes a gap between the two classes of nonlinearities, and e.g. $f(t)=\tfrac{{\rm e}^t}{t+1}$ could not be treated with such techniques. This gap was filled in \cite{Rom_N}, where instead a blow-up approach was used. The same technique has been also applied in \cite{MR} to the respective problem of higher-order, namely when the operator is $(-\Delta)^m$ in $\mathbb R^{2m}$, $m\geq2$, in both cases of Dirichlet and Navier boundary conditions. In these works, the key assumption which allows to deal at once with both classes of nonlinearities in \cite{LRU}, is the existence of
    \begin{equation}\label{b}
        \lim_{t\to+\infty}\frac{f'}f(t)=b\in[0,+\infty)\,
    \end{equation}
    where the case $b>0$ corresponds to the critical setting in the sense of Brezis and Merle, while $b=0$ for all subcritical growths. We point out that with a blow-up approach one may further derive uniform bounds for \textit{weak solutions} of Dirichlet problems with nonlinearity in the class $f(x,u)=u^{p-1}{\rm e}^{u^p}$ even for $p\in(1,2]$, that is, up to the critical Trudinger-Moser growth, under different integral bounds of energy type, see \cite[Theorem 2.2]{DT} and \cite[Theorem 2.1]{MMT}; see also \cite{RobWei,D} for related results.

	When dealing with planar systems, a priori bounds have been investigated in \cite{dFdOR} for systems of the kind \eqref{sys} where $f,g$ are positive nonlinearities (which may also depend on $x$), and then extended also for nonvariational elliptic systems in \cite{dFdOR2}. First, by a moving plane techniques the authors find uniform bounds of the kind $\int_\Omega f(v)\, {\rm d} x\leq\Lambda$ and $\int_\Omega g(u)\, {\rm d} x\leq\Lambda$. Then, in order to apply either Orlicz spaces techniques or the argument inspired by \cite{BM}, again as in \cite{LRU}, a dichotomy in the assumptions was needed: in fact, they impose that $f$ and $g$ grow either less than ${\rm e}^{t^\alpha}$ and ${\rm e}^{t^\beta}$, respectively, with $\alpha,\beta>0$ and $\alpha+\beta<2$, or both like ${\rm e}^t$. In addition to the already recalled gap between the growths ${\rm e}^{t^\alpha}$ and ${\rm e}^t$, the limiting case     \begin{equation}\label{alpha+beta=2}
        \alpha+\beta=2\qquad\mbox{with}\qquad\alpha\neq1\neq\beta
    \end{equation}
    is stated as an open problem in \cite[Remark 1.3]{dFdOR}.

	The main purpose of this article is to progress with the study of a priori bounds for system \eqref{sys} in the conformal setting $N=2$, by covering this case of limiting nonlinearities (which from now on will be called \textit{critical in the sense of Brezis-Merle}), by performing a blow-up analysis in the spirit of \cite{Rom_N,MR}. For the model nonlinearities ${\rm e}^{t^\alpha}$ and ${\rm e}^{t^\beta}$, this will correspond to prove such bounds under \eqref{alpha+beta=2}.  In the companion paper \cite{BMR_sub} we treat instead the subcritical case. We note indeed that such a case is not fully covered by \cite[Theorem 1.3]{dFdOR}, since there are nonlinearities for which $b=0$ but however grow more than any function of the kind ${\rm e}^{pt^\alpha}$ irrespective of the choices of $p\geq1$ and $\alpha\in(0,1)$. For instance, it is sufficient to consider $g(t)={\rm e}^t$ and $f(t)={\rm e}^{\frac t{\log t}}$ so that $\slfrac{f'}f(t)=\frac{\log t-1}{(\log t)^2}\sim\frac1{\log t}\to0$, but slower than any function of the kind ${\rm e}^{pt^\alpha}$. Crucial steps in our methods will be first to identify the right assumption on $f$ and $g$, which detects the ``critical'' nonlinearities and extends \eqref{b} for systems, and next to determine a suitable scaling, that allows to find a limit system in $\mathbb R^2$ to which one may apply a Liouville-type argument. During the blow-up analysis, several technical difficulties will arise due to the non-scalar nature of the problem. In fact, in the proof there will be a continuous exchange of information between the two components of the system. As a first contribution, here we restrict to consider the symmetric case of $\Omega$ being the ball $B_1(0)$: indeed, in this setting we can take advantage to the Schwarz symmetry of the solutions $(u,v)$, which implies that both components cannot have blow-up points except for the origin, and this simplifies to some extent our arguments. However, we believe that with a nontrivial effort one can also apply our strategy even in the case of a general bounded domain of $\mathbb R^2$, also in view of the $L^1$-bounds already proved in \cite{dFdOR}.

    We also mention that blow-up techniques for elliptic systems have been extensively employed in the framework of Toda-type systems,
    where, differently from \eqref{sys}, a strong algebraic structure and integrability properties allow a detailed analysis of bubbling solutions and mass quantization phenomena, see e.g.  \cite{JostWang,LinWeiZhao,LinWeiYangZhang} and references therein. However, the techniques developed in that setting strongly exploit the specific Cartan matrix structure of the  system and therefore do not appear to be directly transferable to the Hamiltonian systems studied here. To the best of our knowledge, we believe that the approach we follow here is new in the literature, and we hope it could serve for other kind of problems.
	\vskip0.2truecm
	Let us now enter into the details of our results.

	\subsection{Assumptions and main results}

	Having in mind the model growths
	\begin{equation}\label{NL_model}
		f(t)\sim{\rm e}^{t^\alpha},\quad g(t)\sim{\rm e}^{t^\beta}\qquad\mbox{with}\ \ \alpha>0,\ \beta>0
	\end{equation}
	with the critical condition
	\begin{equation*}
		\alpha+\beta=2\,,
	\end{equation*}
	we consider the following assumptions:
	\begin{enumerate}
		\item[$(H_1)$] $f,g\in C^1(\overline{\mathbb R^+})$, positive in $\mathbb R^+$, such that $\displaystyle\lim_{t\to+\infty}f(t)=+\infty$ and $\displaystyle\lim_{t\to+\infty}g(t)=+\infty$;
		\item[$(H_2)$] there exist $t_0>0$ such that $f,g$ are strictly increasing in $(t_0,+\infty)$;
        \item[$(H_3)$] $\displaystyle\left(\frac f{f'}\right)'\!\!(t)\to0$ and $\displaystyle\left(\frac{g'}g\right)'\!\!(t)\to0$ as $t\to+\infty$;
		\item[$(H_b)$] there exists $\displaystyle\lim_{t\to+\infty}\,{\frac{f'}f(t)\,\frac{g'}g(t)}:=b\in(0,+\infty)$.
	\end{enumerate}
	Note that $\slfrac{f'}f$ and $\slfrac{g'}g$ are the derivatives of the exponents of $f={\rm e}^{\log f}$ and $g={\rm e}^{\log g}$, respectively. The case $b=0$ corresponds for the model nonlinearities in \eqref{NL_model} to the case $\alpha+\beta<2$, and therefore will be denoted as \textit{Brezis-Merle subcritical}. On the other hand, $\alpha+\beta=2$ implies $b=\alpha\beta>0$, and we will call it \textit{Brezis-Merle critical case}. In this paper we will only deal with the latter case, postponing the analysis of the subcritical one in the companion paper \cite{BMR_sub}, since substantial differences in the respective blow-up arguments will occur.

    In this critical setting, we will also need to distinguish between the case for which the limits of both ratios in $(H_b)$ exist and are finite, from the remaining critical cases. We call the former scenario \textit{Liouville}, which then occurs when there exist $p,q>0$ such that
	\begin{equation}\label{BM}
		\lim_{t\to+\infty}\frac{f'}f(t)=p\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=q\,.
	\end{equation}
	In the latter case, which we call \textit{non-Liouville}, we may instead suppose
	\begin{equation}\label{nonBM}
		\lim_{t\to+\infty}\frac{f'}f(t)=0\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=+\infty
	\end{equation}
	without loss of generality, since the roles of $f$ and $g$ in \eqref{sys} are symmetric. In this last context, we need some monotonicities and growth conditions on $f$ and $g$, 	namely we assume that there exists $t_0>0$ such that
	\begin{enumerate}
        \item[$(H_4)$] $\displaystyle\left(\frac f{f'}\right)'\!\!(t)\geq 0$ and $\displaystyle\left(\frac{g'}g\right)'\!\!(t)\geq 0$ as $t>t_0$;
        \item[$(H_5)$] the map $\displaystyle t\mapsto t\,\frac{f'}f(t)$ is increasing and $\displaystyle t\mapsto\frac1t\,\frac{g'}g(t)$ is decreasing in $(t_0,+\infty)$.
    \end{enumerate}
	Note that the existence of the two limits in \eqref{nonBM} is a consequence of assumptions $(H_b)$ and $(H_4)$. Finally, we suppose one of the following conditions on the growth of $f$: there exists $a>1$ such that
	\begin{enumerate}
		\item[$(H_6)$] $\displaystyle t\,\frac{f'}f(t)\geq a(\log t)^2$ for all $t>t_0$;
		\item[$(H_6')$] $\displaystyle t\,\frac{f'}f(t)\geq a(\log\log F(t))^2$ for all $t>t_0$, where $\displaystyle F(t):=\int_0^tf(\tau)\, {\rm d}\tau$.
	\end{enumerate}

    \vskip0.2truecm
    Before presenting the main result of the paper, let us briefly discuss the assumptions introduced above. First, observe that, except $(H_1)$, are all assumptions on the behaviour at $\infty$. While $(H_1)$ and $(H_2)$ are standard conditions, $(H_3)$ ensures subcriticality in the sense of Trudinger-Moser and that the growth of $f$ and $g$ are superpolynomial (see Lemma \ref{TM-subcritiche} below). The key assumption $(H_b)$, which characterises the problem as critical, can be regarded as the vectorial counterpart of \eqref{b}, see $(A3)$ in \cite{Rom_N} or \cite{MR}, where the scalar case is considered. In the non-Liouville setting, i.e., under condition \eqref{nonBM}, assumptions $(H_4)$ and $(H_5)$ provide suitable monotonicity properties, which are instrumental for our arguments. Note that the existence of the two limits in \eqref{nonBM} is a consequence of the assumptions $(H_b)$ and $(H_4)$. Finally, $(H_6)$ and $(H_6')$ are technical assumptions which imply a mild lower bound on the growth of $f$, which has to be of the kind ${\rm e}^{\log t(\log\log t)^p}$ (see Lemma \ref{lem_f8_conseq} below). Note that, although $(H_6)$ is of easier verification than $(H_6')$, the latter permits to include a larger class of nonlinearities. A more detailed analysis of the consequences of our assumptions can be found in Section \ref{Sec_Prel}.

    \vskip0.2truecm

	Since in the blow-up analysis we will need the radial symmetry of the solutions, as already mentioned in the Introduction, we will only deal with the case of $\Omega$ be a ball in $\mathbb R^2$.
    Our main result is the following:

Let $\Omega\subset\mathbb R^N$ be a smooth bounded domain and consider the semilinear Hamiltonian elliptic system with Dirichlet boundary conditions
 \begin{equation}\label{sys}
 \begin{cases}
 -\Delta u=f(v)\quad&\mbox{in }\Omega\,,\\
 -\Delta v=g(u)\quad&\mbox{in }\Omega\,,\\
 u=v=0\quad&\mbox{on }{\partial\Omega}\,.
 \end{cases}
 \end{equation}
 It is well known that for $N\geq3$ polynomial growth conditions at infinity have to be fulfilled in order to set these variational problems in Sobolev spaces and prove existence.
 If $N\geq3$, for systems of the kind \eqref{sys}, the good notion of criticality is represented by a critical hyperbola: in the model case $f(v)=|v|^{p-1}v$ and $g(u)=|u|^{q-1}u$ with $p,q>1$, then the critical hyperbola defined as
 \begin{equation}\label{hyp}
 \frac1{p+1}+\frac1{q+1}=\frac{N-2}N
 \end{equation}
 divides existence, above \eqref{hyp}, \cite{HV,HMV}, from nonexistence, below \eqref{hyp}, \cite{M}, of classical positive solutions. Note that if \eqref{sys} is set in the entire $\mathbb R^N$ space, then it is proved that the Sobolev hyperbola \eqref{hyp} plays the same role for $N=3,4$, while it is still an open problem, under the name of Lane-Emden conjecture, for dimensions $N\ge 5$, see \cite{M96,PQS,S}. In contrast, much faster growths are admissible in dimension two: in particular, exponential nonlinearities can be treated, with the Trudinger-Moser inequality taking the role of the Sobolev embedding theorem valid for $N\geq3$. For the system \eqref{sys} we still can find a notion of critical hyperbola in the setting of Sobolev-Lorentz spaces: to give an idea, if $f(t)\sim{\rm e}^{|t|^\alpha}$ and $g(t)\sim{\rm e}^{|t|^\beta}$ with $0<\alpha,\beta<+\infty$ (here the symbol $\sim$ means having the same asymptotic growth), then the maximal growth is obtained on the ``conformal'' critical hyperbola
 \begin{equation}\label{abhyp}
 \frac1\alpha+\frac1\beta=1\,,
 \end{equation}
 see \cite{Ruf_sys} and the more recent advance in \cite{dORR}.

Having in mind the model growths
 \begin{equation}\label{NL_model}
 f(t)\sim{\rm e}^{t^\alpha},\quad g(t)\sim{\rm e}^{t^\beta}\qquad\mbox{with}\ \ \alpha>0,\ \beta>0
 \end{equation}
 with the critical condition
 \begin{equation*}
 \alpha+\beta=2\,,
 \end{equation*}
 we consider the following assumptions:
 \begin{enumerate}
 \item[$(H_1)$] $f,g\in C^1(\overline{\mathbb R^+})$, positive in $\mathbb R^+$, such that $\displaystyle\lim_{t\to+\infty}f(t)=+\infty$ and $\displaystyle\lim_{t\to+\infty}g(t)=+\infty$;
 \item[$(H_2)$] there exist $t_0>0$ such that $f,g$ are strictly increasing in $(t_0,+\infty)$;
 \item[$(H_3)$] $\displaystyle\left(\frac f{f'}\right)'\!\!(t)\to0$ and $\displaystyle\left(\frac{g'}g\right)'\!\!(t)\to0$ as $t\to+\infty$;
 \item[$(H_b)$] there exists $\displaystyle\lim_{t\to+\infty}\,{\frac{f'}f(t)\,\frac{g'}g(t)}:=b\in(0,+\infty)$.
 \end{enumerate}
 Note that $\slfrac{f'}f$ and $\slfrac{g'}g$ are the derivatives of the exponents of $f={\rm e}^{\log f}$ and $g={\rm e}^{\log g}$, respectively. The case $b=0$ corresponds for the model nonlinearities in \eqref{NL_model} to the case $\alpha+\beta<2$, and therefore will be denoted as \textit{Brezis-Merle subcritical}. On the other hand, $\alpha+\beta=2$ implies $b=\alpha\beta>0$, and we will call it \textit{Brezis-Merle critical case}. In this paper we will only deal with the latter case, postponing the analysis of the subcritical one in the companion paper \cite{BMR_sub}, since substantial differences in the respective blow-up arguments will occur.

In this critical setting, we will also need to distinguish between the case for which the limits of both ratios in $(H_b)$ exist and are finite, from the remaining critical cases. We call the former scenario \textit{Liouville}, which then occurs when there exist $p,q>0$ such that
 \begin{equation}\label{BM}
 \lim_{t\to+\infty}\frac{f'}f(t)=p\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=q\,.
 \end{equation}
 In the latter case, which we call \textit{non-Liouville}, we may instead suppose
 \begin{equation}\label{nonBM}
 \lim_{t\to+\infty}\frac{f'}f(t)=0\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=+\infty
 \end{equation}
 without loss of generality, since the roles of $f$ and $g$ in \eqref{sys} are symmetric. In this last context, we need some monotonicities and growth conditions on $f$ and $g$, namely we assume that there exists $t_0>0$ such that
 \begin{enumerate}
 \item[$(H_4)$] $\displaystyle\left(\frac f{f'}\right)'\!\!(t)\geq 0$ and $\displaystyle\left(\frac{g'}g\right)'\!\!(t)\geq 0$ as $t>t_0$;
 \item[$(H_5)$] the map $\displaystyle t\mapsto t\,\frac{f'}f(t)$ is increasing and $\displaystyle t\mapsto\frac1t\,\frac{g'}g(t)$ is decreasing in $(t_0,+\infty)$.
 \end{enumerate}
 Note that the existence of the two limits in \eqref{nonBM} is a consequence of assumptions $(H_b)$ and $(H_4)$. Finally, we suppose one of the following conditions on the growth of $f$: there exists $a>1$ such that
 \begin{enumerate}
 \item[$(H_6)$] $\displaystyle t\,\frac{f'}f(t)\geq a(\log t)^2$ for all $t>t_0$;
 \item[$(H_6')$] $\displaystyle t\,\frac{f'}f(t)\geq a(\log\log F(t))^2$ for all $t>t_0$, where $\displaystyle F(t):=\int_0^tf(\tau)\, {\rm d}\tau$.
 \end{enumerate}

\vskip0.2truecm

Since in the blow-up analysis we will need the radial symmetry of the solutions, as already mentioned in the Introduction, we will only deal with the case of $\Omega$ be a ball in $\mathbb R^2$.
 Our main result is the following:

\vskip0.2truecm

Note that, under such growth assumptions, one can easily show that $f$ grows less than or comparable to $\e^{pt}$ for some $p>0$ (see Lemma \ref{growth_below_g} below). Hence we may apply the regularity result \cite[Theorem 1.1]{dFdOR} and conclude that distributional solutions of \eqref{sys} are indeed classical. Therefore, from now on we will simply talk about \textit{solutions of \eqref{sys}}.

We follow the approach of \cite{QS}, which is based on the \textit{Fixed Point} (or Leray-Schauder) Index theory, and we refer to \cite{AdF} for its definition and basic properties. In what follows in view of Theorem \ref{Thm_Existence} we consider $\Omega=B_1(0)$, however, once such a priori bound is recovered in a smooth bounded domain $\Omega\subset\R^2$, the proof of Theorem \ref{Thm_Uniform_bound} works without changes.
 \vskip0.2truecm
 Let $K$ be the cone of functions in $C(\Omegabar)\times C(\Omegabar)$ which are positive in $\Omega$ and denote by $S$ the solution operator of the system
 \begin{equation*}
 \begin{cases}
 -\Delta u=\phi\quad&\mbox{in }\Omega\,,\\
 -\Delta v=\psi\quad&\mbox{in }\Omega\,,\\
 u=v=0\quad&\mbox{on }\dOmega\,,
 \end{cases}
 \end{equation*}
 namely $S(\phi,\psi)=(u,v)$. Note that $S$ is a linear compact operator from $C(\Omegabar)\times C(\Omegabar)$ into itself. Hence solutions of \eqref{sys} correspond to fixed points of the map $T:(u,v)\mapsto T(u,v):=S(f(v),g(u))$, which is compact by $(H_1)$.
 For $W\subset K$ relatively open such that $Tz\neq z$ for all $z\in\partial W$ we denote by $i_K(T,W)$ the
 fixed point index of $T$ in $W$ with respect to $K$. We are going to prove that there exist $0<r<R$ such that $i_K(T,W_R\setminus\overline{W_r})\neq0$, by showing that $i_K(T,W_r)=1$ while $i_K(T,W_R)=0$. Here $W_s:=B_s(0)\cap K$, where $B_s(0)$ for $s>0$ denotes the ball in $C(\Omegabar)\times C(\Omegabar)$ of radius $s$ and centred at $0$. To prove that there exists $r$ small such that the index in $W_r$ equals $1$, we make use of the superlinearity of $f$ and $g$ at $0$; instead, to prove the index vanishes in $W_R$ for a large $R$, we exploit the a priori bound by Theorem \ref{Thm_Uniform_bound} together with the superlinearity of the nonlinearities at $\infty$.
 \vskip0.2truecm
 Let us introduce the homothety $H_1(t,u,v)=tT(u,v)$ and prove that there exists $r>0$ such that $H_1(t,u,v)\neq(u,v)$ for all $t\in[0,1]$ if $(u,v)\in\partial W_r$. To this aim fix $\varepsilon\in(0,\lambda_1)$ and by $(H_9)$ take $r>0$ such that if $\|u\|_\infty+\|v\|_\infty=r$, one has $f(v)\leq\varepsilon v$ and $g(u)\leq\varepsilon u$. Introduce $\phi_1$ as the first (positive) eigenfunction of $-\Delta$ in $\Omega$. Supposing that there exists $(u,v)$ such that $H_1(t,u,v)=(u,v)$, namely a solution of
 \begin{equation*}
 \begin{cases}
 -\Delta u=tf(v)\quad&\mbox{in }\Omega\,,\\
 -\Delta v=tg(u)\quad&\mbox{in }\Omega\,,\\
 u=v=0\quad&\mbox{on }\dOmega\,,
 \end{cases}
 \end{equation*}
 we test the system with $\phi_1$ and sum the two equations:
 \begin{equation*}
 \lambda_1\intOmega(u+v)\phi_1=\intOmega(\nabla u+\nabla v)\nabla \phi_1=t\intOmega\left(f(v)+g(u)\right)\phi_1\leq\varepsilon t\intOmega(u+v)\phi_1\,,
 \end{equation*}
 which is a contradiction since $t\in[0,1]$ and $\varepsilon<\lambda_1$. As a consequence,
 \begin{equation}\label{Index_1}
 i_K(T,W_r)=i_K\left(H_1(1,\cdot,\cdot),W_r\right)=i_K\left(H_1(0,\cdot,\cdot),W_r\right)=i_K(0,W_r)=1\,.
 \end{equation}

Define now a second homothety $H_2(\mu,u,v)=S( f(v+\mu),g(u+\mu))$ where $\mu\ge 0$, the fixed point of which are solutions of
 \begin{equation}\label{sys_mu}
 \begin{cases}
 -\Delta u=f(v+\mu)\quad&\mbox{in }\Omega\,,\\
 -\Delta v=g(u+\mu)\quad&\mbox{in }\Omega\,,\\
 u=v=0\quad&\mbox{on }\dOmega\,.
 \end{cases}
 \end{equation}
 By $(H_3)$ we know that both $f$ and $g$ are superlinear (see Lemma \ref{TM-subcritiche}), that is, for any $M>0$ there exists $t_0>0$ such that for all $t>t_0$ one has $f(t)>Mt$ and $g(t)>Mt$. Again by testing the system with $\phi_1$, we obtain
 \begin{equation*}
 \begin{split}
 \intOmega f(v+\mu)\phi_1&=\intOmega\nabla u\nabla\phi_1=\lambda_1\left(\int_{\{u\leq t_0\}}u\phi_1+\int_{\{u>t_0\}}u\phi_1\right)\\
 &\leq\lambda_1\left(t_0\|\phi_1\|_1+\frac1M\intOmega g(u+\mu)\phi_1\right),
 \end{split}
 \end{equation*}
 since $\mu\geq0$ and $\phi_1>0$. Similarly
 \begin{equation*}
 \intOmega g(u+\mu)\phi_1=\intOmega\nabla v\nabla\phi_1\leq\lambda_1\left(t_0\|\phi_1\|_1+\frac1M\intOmega f(v+\mu)\phi_1\right).
 \end{equation*}
 These together yield
 \begin{equation*}
 \left(1-\frac{\lambda_1^2}{M^2}\right)\intOmega f(v+\mu)\phi_1\leq\lambda_1\left(1+\frac{\lambda_1}M\right)t_0\|\phi_1\|_1\,.
 \end{equation*}
 and
 \begin{equation*}
 \left(1-\frac{\lambda_1^2}{M^2}\right)\intOmega g(u+\mu)\phi_1\leq\lambda_1\left(1+\frac{\lambda_1}M\right)t_0\|\phi_1\|_1\,.
 \end{equation*}
 Choosing $M=2\lambda_1$, one then finds
 \begin{equation}\label{Bound_l1loc}
 \intOmega f(v+\mu)\phi_1\leq2\lambda_1t_0\|\phi_1\|_1\quad\ \mbox{and}\ \quad\intOmega g(u+\mu)\phi_1\leq2\lambda_1t_0\|\phi_1\|_1\,,
 \end{equation}
 which by $(H_1)$ and $u,v\geq0$ imply that $\mu$ needs to be bounded from above. Hence, there exists $\overline\mu>0$ such that for all $\mu>\overline\mu$ one has $H_2(\mu,u,v)\neq(u,v)$ in $K$. Moreover, for any fixed $\mu_0$, we note that $f(\cdot+\mu)$ and $g(\cdot+\mu)$ satisfy the assumptions $(H_1)$-$(H_6')$ uniformly for $\mu\in[0,\mu_0]$. In fact, it is evident that $(H_1)$-$(H_4)$ hold; moreover, $\frac1t\frac f{f'}(t+\mu)=\left(1+\frac\mu t\right)\frac1{t+\mu}\frac f{f'}(t+\mu)$, which is decreasing for large $t$ and uniformly in $\mu\in[0,\mu_0]$. Indeed, this is a product of two eventually decreasing positive functions by $(H_5)$ applied to $f$; the same argument applies also for $g$, hence $(H_5)$ holds also for $f(\cdot+\mu)$ and $g(\cdot+\mu)$. Similarly, one also verifies $(H_6)$ and $(H_6')$: indeed, e.g.
 $$\frac t{(\log t)^2}\frac{f'}f(t+\mu)=\frac{t(\log(t+\mu))^2}{(t+\mu)(\log t)^2}\cdot\frac{(t+\mu)}{(\log(t+\mu))^2}\frac{f'}f(t+\mu)\geq\frac a2$$
 for $t$ large, where $a$ is the constant in $(H_6)$ for $\mu=0$, since the first term converges to $1$ as $t\to+\infty$ uniformly in $\mu\in[0,\mu_0]$. Moreover, it is important to note that the constant $\Lambda$ in Proposition \ref{Prop_Lambda} is independent of $\mu$, provided $\mu\in[0,\mu_0]$. Indeed, inspecting its proof in \cite[Theorem 1.2]{dFdOR}, it is sufficient that \eqref{Bound_l1loc} holds with a bound independent of $\mu\in[0,\mu_0]$ to conclude. After all these considerations, we can thus state that Theorem \ref{Thm_Uniform_bound} applies with a bound uniform with respect to $\mu\in[0,\mu_0]$. Therefore taking $\mu_0=\overline\mu+1>0$ there exists $\overline R=\overline R(\overline\mu)>0$ such that all solutions of \eqref{sys_mu} are bounded in $L^\infty$ norm by $\overline R/2$. Hence, $H_2(\mu,u,v)\neq(u,v)$ for all $(u,v)\in B_{\overline R}(0)^c\cap K$ and $\mu\in[0,\overline\mu+1]$. Hence, combining the two information on the map $H_2$, we infer
 \begin{equation}\label{Index_0}
 i_K(T,W_{\overline R})=i_K\left(H_2(0,\cdot,\cdot),W_{\overline R}\right)=i_K\left(H_2(\overline\mu+1,\cdot,\cdot),W_{\overline R}\right)=i_K(0,W_{\overline R})=0\,.
 \end{equation}
 As a result, from \eqref{Index_0} and \eqref{Index_1} one deduces $i_K(T,W_R\setminus\overline{W_r})=-1\neq0$, which yields the existence of a nontrivial positive solution of \eqref{sys}.

Note that, under such growth assumptions, one can easily show that $f$ grows less than or comparable to ${\rm e}^{pt}$ for some $p>0$ (see Lemma \ref{growth_below_g} below). Hence we may apply the regularity result \cite[Theorem 1.1]{dFdOR} and conclude that distributional solutions of \eqref{sys} are indeed classical. Therefore, from now on we will simply talk about \textit{solutions of \eqref{sys}}.

    \begin{figure}[h!]
    \centering
    \begin{tikzpicture}
    \begin{axis}[
        axis lines=none,
        xmin=0, xmax=3.5,
        ymin=0, ymax=3.5,
        width=8cm, height=8cm,
        clip=false
    ]
    \addplot [name path=retta, domain=0:2, samples=2, draw=none] {2-x};
    \addplot [name path=assex, domain=0:2, samples=2, draw=none] {0};
    \addplot [fill=blue!15, draw=none] fill between[of=retta and assex, soft clip={domain=0:2}];
    \end{axis}

    \begin{axis}[
        axis lines=middle,
        xlabel={$\alpha$},
        ylabel={$\beta$},
        xmin=0, xmax=3.5,
        ymin=0, ymax=3.5,
        xtick={0,1,2,3},
        ytick={0,1,2,3},
        xticklabels={0,1,2,3},
        yticklabels={0,1,2,3},
        axis line style={->},
        width=8cm, height=8cm,
        clip=false,
        label style={font=\small},
        axis on top
    ]

    \addplot[red!80!black, very thick, domain=0:2, samples=2] {2 - x};

    \addplot[
        only marks,
        mark=o,
        mark size=2pt,
        mark options={draw=blue!80!black, line width=1pt}
    ] coordinates {(1,1)};

    \addplot[green!50!black, dashed, ultra thick, samples=200, domain=1.01:3.5, restrict y to domain=0:3.5]
        {1/(1 - 1/x)};

    \node[below left] at (axis cs:0,0) {\small 0};

    \end{axis}
    \end{tikzpicture}
    \caption{The state of the art on uniform a priori bounds for \eqref{sys} for the model case \eqref{NL_model}.}
    \label{fig1}
    \end{figure}

    In Figure \ref{fig1} we can see the state of the art for problem \eqref{sys} for the model nonlinearities in \eqref{NL_model}. In particular, a priori bounds in the blue area and in the point $(1,1)$ were proved in \cite{dFdOR}, the red line is our contribution, while the dashed curve in green is the ``conformal'' critical hyperbola \eqref{abhyp}.
    This picture underlines that our case is largely subcritical in the sense of Trudinger-Moser.  Note that we retrieve the scalar results in \cite{LRU,Rom_N} if we restrict on the diagonal of the graph. Therefore, in light of the counterexamples in \cite{BM,MR,Rom_N}, we expect that the region where now the a priori bounds for \eqref{sys} applies is optimal for \textit{distributional} solutions at least in the presence of singular potentials. However, it is still not clear -- even in the scalar case -- whether a priori bounds for \textit{weak} solutions may be proved up to the critical growth in the sense of Trudinger-Moser.

    \vskip0.2truecm
	Although reminiscent of some ideas developed in \cite{MR,Rom_N} to deal with corresponding scalar problems, the method that we here present is new and completely different from the one in \cite{dFdOR}, from which we take only the a priori $L^1$ bounds in Proposition \ref{Prop_Lambda}. Our blow-up technique is based on a suitable new scaling for system \eqref{sys}, and the main challenge is to precisely understand how to profit from the exchange of information between the two components $u$ and $v$ in order to find contradictions with such $L^1$-bounds. Let us try to briefly sketch the main line of our argument. We start in Section \ref{Sec_scaling} by assuming by contradiction that there exists a sequence of solutions $(u_k,v_k)_k$ of \eqref{sys} such that
    $$
    \max\{\|u_k\|_\infty,\|v_k\|_\infty\}=+\infty\,.
    $$

    First, we exclude the possibility that only one among $(\|u_k\|_\infty)_k$ and $(\|v_k\|_\infty)_k$ is unbounded and, exploiting the symmetric setting, we easily see that the origin is the unique blow-up point for both sequences.

    The goal is to reach a contradiction either with the uniform $L^1$-bounds on the nonlinear terms (see Proposition \ref{Prop_Lambda}) or by applying a suitable Liouville-type theorem in the plane. In both cases, detecting a suitable scaling is essential: it produces $(\tu_k,\tv_k)_k$ defined on balls $\Omega_k$ with diverging radii, which formally locally converges to a solution of a Liouville's system on $\mathbb R^2$. In order to ensure this, we need that the scalings for both components are coupled by a compatibility condition, which we analyse in detail in Section \ref{sec:compatibility}.

    Choosing to base the scaling on $\|u_k\|_\infty=u_k(0)$, so that $\tu_k(0)=0$, our blow-up analysis in Section \ref{Sec_blowup} then proceeds by distinguishing several cases according to the behavior of $\tv_k(0)$ as $k\to\infty$. If $\tv_k(0)\to-\infty$ as $k\to\infty$, then the Harnack inequality can be directly applied to $\tu_k$, which leads to a simple contradiction with the energy bounds. If $\tv_k(0)\to c\in\mathbb R$, then in the non-Liouville case \eqref{nonBM} the boundedness of $\tv_k$ on compact sets yields a similar situation. In the Liouville case \eqref{BM}, we infer the contradiction from the analysis of the limit Liouville's system, relying on a characterisation result in \cite{CK}. Finally, the analysis becomes more technical when $\tv_k(0)\to +\infty$ and we need to find finer global estimates on $\tu_k$ and $\tv_k$, especially in the non-Liouville setting. Here the additional assumptions $(H_6)$ and $(H_6')$ will permit to find the desired contradiction.

	\vskip0.2truecm

	Under a standard superlinear growth condition on the nonlinearities in $0$, in Section \ref{Sec_Existence} we show that once the a priori bound is obtained, then the existence of a positive continuous solution $(u,v)$ of \eqref{sys} follows by means of Fixed Point index theory.

    \begin{thm}\label{Thm_Existence}
        Besides the conditions under which Theorem \ref{Thm_Uniform_bound} holds, assume moreover
        \begin{enumerate}
            \item[$(H_9)$] $\displaystyle\limsup_{t\to0^+}\frac{f(t)}t<\lambda_1\ $ and $\ \,\displaystyle\limsup_{t\to0^+}\frac{g(t)}t<\lambda_1\,$,
    	\end{enumerate}
        where $\lambda_1$ is the first eigenvalue of $-\Delta$ in $\Omega$ with Dirichlet boundary conditions. Then \eqref{sys} admits a positive solution.
    \end{thm}

	\paragraph{\textbf{Notation.}} For $R>0$ and $x_0\in\mathbb R^N$ we denote by $B_R(x_0)$ the ball of radius $R$ and center $x_0$. The symbol $o_n(1)$ denotes a vanishing real sequence as $n\to+\infty$. Hereafter, the letter $C$ will be used to denote positive constants which are independent of relevant quantities and whose value may change from line to line.

To prove Theorem~\ref{Thm_Uniform_bound}, the authors “assum[e] by contradiction that there exists a sequence of solutions $(u_k,v_k)_k$ of \eqref{sys} such that
\[
\max\{\|u_k\|_\infty,\|v_k\|_\infty\}=+\infty\,.
\]”
First, they “exclude the possibility that only one among $(\|u_k\|_\infty)_k$ and $(\|v_k\|_\infty)_k$ is unbounded” and, using the symmetry, show that “the origin is the unique blow-up point for both sequences.”

The aim is “to reach a contradiction either with the uniform $L^1$-bounds on the nonlinear terms (see Proposition~\ref{Prop_Lambda}) or by applying a suitable Liouville-type theorem in the plane.” A key ingredient is “detecting a suitable scaling”: it produces rescaled functions $(\tilde u_k,\tilde v_k)_k$ on balls $\Omega_k$ “with diverging radii,” which “formally locally converges to a solution of a Liouville's system on $\mathbb R^2$.” To make this work, “we need that the scalings for both components are coupled by a compatibility condition,” analyzed in Section~\ref{sec:compatibility}.

They “base the scaling on $\|u_k\|_\infty=u_k(0)$, so that $\tilde u_k(0)=0$,” and then distinguish cases depending on the behavior of $\tilde v_k(0)$ as $k\to\infty$:
\begin{itemize}
\item If $\tilde v_k(0)\to-\infty$, “the Harnack inequality can be directly applied to $\tilde u_k$,” giving “a simple contradiction with the energy bounds.”
\item If $\tilde v_k(0)\to c\in\mathbb R$, then in the non-Liouville case \eqref{nonBM} the “boundedness of $\tilde v_k$ on compact sets yields a similar situation,” while in the Liouville case \eqref{BM} they get a contradiction from “the analysis of the limit Liouville's system,” using “a characterisation result in \cite{CK}.”
\item If $\tilde v_k(0)\to+\infty$, the argument is “more technical” and requires “finer global estimates on $\tilde u_k$ and $\tilde v_k$, especially in the non-Liouville setting”; the extra assumptions $(H_6)$ or $(H_6')$ “permit to find the desired contradiction.”
\end{itemize}
In all cases, this yields the contradiction needed to conclude the uniform a priori bounds of Theorem~\ref{Thm_Uniform_bound}.

To prove the main theorem, the authors “assum[e] by contradiction that there exists a sequence of solutions $(u_k,v_k)_k$ of
\begin{equation}\label{sys}
		\begin{cases}
			-\Delta u=f(v)\quad&\mbox{in }\Omega\,,\\
			-\Delta v=g(u)\quad&\mbox{in }\Omega\,,\\
			u=v=0\quad&\mbox{on }\dOmega\,.
		\end{cases}
	\end{equation}
 such that
\[
\max\{\|u_k\|_\infty,\|v_k\|_\infty\}=+\infty\,.
\]”
First, they “exclude the possibility that only one among $(\|u_k\|_\infty)_k$ and $(\|v_k\|_\infty)_k$ is unbounded” and, using the symmetry, show that “the origin is the unique blow-up point for both sequences.”

The aim is “to reach a contradiction either with the uniform $L^1$-bounds on the nonlinear terms,” namely the following result:
\begin{prop}[Theorem 1.2, \cite{dFdOR}]\label{Prop_Lambda}
		Assume that there exists $c>0$ and $p>0$ such that $f(t)\leq c\e^{pt}$ for all $t>0$, or that the same holds for $g$. Then there exists a positive constant $\Lambda$, depending only on $f$, $g$, and $\Omega$, such that
        \begin{equation}\label{Lambda}\tag{$\Lambda$}
            \intOmega f(v)\leq\Lambda\quad\mbox{and}\quad\intOmega g(u)\leq\Lambda
        \end{equation}
        for all solutions of \eqref{sys}.
	\end{prop}
or by applying a suitable Liouville-type theorem in the plane.” A key ingredient is “detecting a suitable scaling”: it produces rescaled functions $(\tilde u_k,\tilde v_k)_k$ on balls $\Omega_k$ “with diverging radii,” which “formally locally converges to a solution of a Liouville's system on $\mathbb R^2$.” To make this work, “we need that the scalings for both components are coupled by a compatibility condition,” analyzed later.

They “base the scaling on $\|u_k\|_\infty=u_k(0)$, so that $\tilde u_k(0)=0$,” and then distinguish cases depending on the behavior of $\tilde v_k(0)$ as $k\to\infty$:
\begin{itemize}
\item If $\tilde v_k(0)\to-\infty$, “the Harnack inequality can be directly applied to $\tilde u_k$,” giving “a simple contradiction with the energy bounds.”
\item If $\tilde v_k(0)\to c\in\mathbb R$, then in the non-Liouville case
\begin{equation}\label{nonBM}
		\lim_{t\to+\infty}\frac{f'}f(t)=0\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=+\infty
	\end{equation}
the “boundedness of $\tilde v_k$ on compact sets yields a similar situation,” while in the Liouville case
\begin{equation}\label{BM}
		\lim_{t\to+\infty}\frac{f'}f(t)=p\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=q\,.
	\end{equation}
they get a contradiction from “the analysis of the limit Liouville's system,” using “a characterisation result in \cite{CK}.”
\item If $\tilde v_k(0)\to+\infty$, the argument is “more technical” and requires “finer global estimates on $\tilde u_k$ and $\tilde v_k$, especially in the non-Liouville setting”; the extra assumptions $(H_6)$ or $(H_6')$ “permit to find the desired contradiction.”
\end{itemize}
In all cases, this yields the contradiction needed to conclude the uniform a priori bounds of the main theorem.

\label{Thm_Uniform_bound}
		Let $\Omega=B_1(0)$ be the unit ball in $\mathbb R^2$, assume conditions $(H_1)$, $(H_2)$, $(H_3)$, $(H_b)$, and alternatively
		\begin{enumerate}
			\item[i)] \begin{equation}\label{BM}
		\lim_{t\to+\infty}\frac{f'}f(t)=p\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=q\,.
	\end{equation} holds;
			\item[ii)] \begin{equation}\label{nonBM}
		\lim_{t\to+\infty}\frac{f'}f(t)=0\quad\mbox{and}\quad\lim_{t\to+\infty}\frac{g'}g(t)=+\infty
	\end{equation}, $(H_4)$, $(H_5)$, and either $(H_6)$ or $(H_6')$ hold.
		\end{enumerate}
		Then there exists a constant $C>0$ such that
		\begin{equation*}
			\|u\|_\infty\leq C\qquad\mbox{and}\qquad\|v\|_\infty\leq C
		\end{equation*}
		for all (eventual) solutions $(u,v)$ of the system
		\begin{equation}\label{sys}
		\begin{cases}
			-\Delta u=f(v)\quad&\mbox{in }\Omega\,,\\
			-\Delta v=g(u)\quad&\mbox{in }\Omega\,,\\
			u=v=0\quad&\mbox{on }\dOmega\,.
		\end{cases}
	\end{equation}.

Question:
Let \(\Omega=B_1(0)\subset\mathbb R^2\) be the unit ball, and consider the Dirichlet system
\[
\begin{cases}
-\Delta u=f(v) & \text{in }\Omega,\\
-\Delta v=g(u) & \text{in }\Omega,\\
u=v=0 & \text{on }\partial\Omega.
\end{cases}
\]
Assume \(f,g\in C^1([0,\infty))\), \(f(t),g(t)>0\) for \(t>0\), \(f(t)\to\infty\) and \(g(t)\to\infty\) as \(t\to\infty\), and that there exists \(t_0>0\) such that \(f\) and \(g\) are strictly increasing on \((t_0,\infty)\). Assume also
\[
\left(\frac{f}{f'}\right)'(t)\to 0,
\qquad
\left(\frac{g'}{g}\right)'(t)\to 0
\quad \text{as } t\to\infty,
\]
and
\[
\lim_{t\to\infty}\frac{f'(t)}{f(t)}\frac{g'(t)}{g(t)}=b\in(0,\infty).
\]
Suppose, moreover, that either
1. (Liouville case) there exist \(p,q>0\) such that
\[
\lim_{t\to\infty}\frac{f'(t)}{f(t)}=p,
\qquad
\lim_{t\to\infty}\frac{g'(t)}{g(t)}=q,
\]
or
2. (non-Liouville case)
\[
\lim_{t\to\infty}\frac{f'(t)}{f(t)}=0,
\qquad
\lim_{t\to\infty}\frac{g'(t)}{g(t)}=+\infty,
\]
and there exists \(t_0>0\) such that
\[
\left(\frac{f}{f'}\right)'(t)\ge 0,
\qquad
\left(\frac{g'}{g}\right)'(t)\ge 0
\quad \text{for } t>t_0,
\]
with \(t\mapsto t\,\frac{f'(t)}{f(t)}\) increasing and \(t\mapsto \frac1t\,\frac{g'(t)}{g(t)}\) decreasing on \((t_0,\infty)\), and in addition one of the further growth assumptions \((H_6)\) or \((H_6')\) holds.

Under these assumptions, which quantitative estimate holds for all solutions \((u,v)\) of the system?

There exists a constant \(C>0\) such that every solution \((u,v)\) satisfies \(\|u\|_{\infty}\le C\) and \(\|v\|_{\infty}\le C\).

Choices:
B. There exists a constant \(C>0\), depending only on \(f\), \(g\), and \(\Omega\), such that every solution \((u,v)\) satisfies \(\|u\|_{L^1(\Omega)}\le C\) and \(\|v\|_{L^1(\Omega)}\le C\).
C. For every solution \((u,v)\), both components are bounded, i.e. \(u,v\in L^{\infty}(\Omega)\).
D. There exists a constant \(C>0\) such that every radial solution \((u,v)\) satisfies \(\|u\|_{\infty}\le C\) and \(\|v\|_{\infty}\le C\).
E. There exists a constant \(C>0\) such that every solution \((u,v)\) satisfies \(\int_{\Omega} f(v)\,dx\le C\) and \(\int_{\Omega} g(u)\,dx\le C\), and consequently at least one of \(\|u\|_{\infty}\) or \(\|v\|_{\infty}\) is bounded by \(C\).

replaces uniform sup-norm conclusion by only componentwise L1 bounds

drops the existence of a single uniform constant valid for all solutions

restricts the conclusion from all solutions to radial solutions only

misuses the uniform integral bounds on nonlinear terms as a substitute for two-sided uniform L-infinity control


If a computably axiomatized theory~$T_0$ is a conservative extension of another such theory~$T_1$ for a syntactical class of sentences $\Gamma$, there is an algorithmic procedure which translates any $T_0$-proof~$p$ of a $\Gamma$-sentence into its shortest $T_1$-proof~$\hat p$. It is natural to wonder whether the translation $p \mapsto \hat p$ yields significantly longer proofs. If there exists a $T_0$-proof~$p$ of a $\Gamma$-sentence such that the length $|\hat p|$ of its translated $T_1$-proof is super-polynomial with respect to the length~$|p|$, we say that $T_0$ admits \emph{non-trivial speedup} over $T_1$ for $\Gamma$-formulas. In this case, the theory~$T_0$ is arguably useful for $\Gamma$-sentences, in that it sometimes produces significantly shorter proofs than~$T_1$ for such sentences. If on the other hand, there exists a polynomial~$Q$ such that $|\hat p| \leq Q(|p|)$ for every $T_0$-proof~$p$ of any $\Gamma$-sentence, then we consider that $T_0$ admits no significant speedup over~$T_1$ for $\Gamma$-formulas, since the various proof systems are mutually polynomially simulated.

The study of proof size and proof speedups traces back to G\"odel~\cite{davis1990kurt} (see Pudl\'ak~\cite{pudlak1998length} for an excellent survey). However, it was only in the 1990's that Avigad~\cite{avigad1996formalizing} showed that model-theoretic conservation theorems based on forcing could be formalized into proof-theoretic arguments and yield proof size analysis. He proved in particular that $\WKL_0$ is $\Pi^1_1$-conservative over $\RCA_0$ with no significant increase in the length of proofs. The technique was later used to formalize $\Pi^1_1$-conservation theorems over~$\RCA_0+\ISig_n$~\cite{Ikari-PhD,Ik-Yo} and $\forall \Sigma^0_2$-conservation theorems over~$\RCA_0$~\cite{kolodziejczyk2023ramsey} and over $\RCA_0^*$~\cite{katarzyna2025speedup}, where a $\forall \Sigma^0_n$-formula is of the form $\forall X \varphi(X)$ where $\varphi$ is $\Sigma^0_n$.

We are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\RT^2_2$). Patey and Yokoyama~\cite{patey2018proof} proved that $\WKL_0 + \RT^2_2$ is $\forall \Sigma^0_2$-conservative over~$\RCA_0$ using the notion of $\alpha$-largeness from Ketonen and Solovay~\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\l}odziejczyk and Yokoyama~\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\l}odziejczyk, Wong and Yokoyama~\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\WKL_0^* + \RT^2_2$ yields a non-elementary speedup over~$\RCA_0^*$ even for $\Sigma_1$-sentences~\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\WKL_0 + \RT^2_2$ is $\forall \Sigma^0_3$-conservative over $\RCA_0 + \BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.

In this article, we define a variant of the parameterized version of $\alpha$-largeness \cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\cite{avigad1996formalizing} and developed in the context of $\alpha$-largeness by Ko{\l}odziejczyk, Wong and Yokoyama~\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\WKL_0 + \RT^2_2$ over $\RCA_0 + \BSig_2$ for $\forall \Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:

We are particularly interested in the first-order consequences of Ramsey's theorem for pairs and two colors ($\RT^2_2$). Patey and Yokoyama~\cite{patey2018proof} proved that $\WKL_0 + \RT^2_2$ is $\forall \Sigma^0_2$-conservative over~$\RCA_0$ using the notion of $\alpha$-largeness from Ketonen and Solovay~\cite{ketonen1981rapidly}. The proof was later simplified by Ko{\l}odziejczyk and Yokoyama~\cite{kolo2020some} to give explicit polynomial bounds for this notion of largeness. Ko{\l}odziejczyk, Wong and Yokoyama~\cite{kolodziejczyk2023ramsey} then formalized the construction to prove that the conservation proof does not increase significantly the length of proofs. This constrasts with the fact that $\WKL_0^* + \RT^2_2$ yields a non-elementary speedup over~$\RCA_0^*$ even for $\Sigma_1$-sentences~\cite{kolodziejczyk2023ramsey}. More recently, Le Houérou, Levy Patey and Yokoyama~\cite{houerou2026pi} defined a parameterized version of Ketonen and Solovay's notion of largeness and proved that $\WKL_0 + \RT^2_2$ is $\forall \Sigma^0_3$-conservative over $\RCA_0 + \BSig_2$, with explicit bounds computation. However, these bounds are exponential, leaving open whether the conservation theorem yields a significant proof speedup.

In this article, we define a variant of the parameterized version of $\alpha$-largeness \cite{houerou2026pi} and prove that Ramsey's theorem for pairs admits polynomial bounds with respect to this notion of largeness. Then, using the now-called notion of forcing interpretation introduced by Avigad~\cite{avigad1996formalizing} and developed in the context of $\alpha$-largeness by Ko{\l}odziejczyk, Wong and Yokoyama~\cite{kolodziejczyk2023ramsey}, we prove that the conservation theorem for $\WKL_0 + \RT^2_2$ over $\RCA_0 + \BSig_2$ for $\forall \Sigma^0_3$-sentences does not yield any significant proof speedup. The polynomial bounds computation for this new notion of largeness is non-trivial, and require to prove a partition theorem for trees based on a finite version of Milliken's tree theorem~\cite{milliken1979ramsey,dodos2016ramsey} with primitive recursive bounds. Our main theorem is therefore the following:

The article is divided into three parts : in \Cref{sec:largeness}, we survey and compare three related notions of largeness, namely, Ketonen and Solvay's $\alpha$-largeness, Le Houérou, Levy Patey and Yokoyama's $\alpha$-largeness$(\theta)$, and our new notion of $\alpha$-largeness${}^*(\theta)$, for $\alpha$ of the form $\bbomega^n \cdot k$. In \Cref{sec:combinatorics}, we develop the framework of $\alpha$-largeness${}^*(\theta)$ and prove polynomial upper bounds for $\RT^2_2$-$\bbomega^n$-largeness${}^*(\theta)$. Last, in \Cref{sec:translation}, we combine this upper bound with the framework of forcing interpretation to prove \Cref{thm:rt22-polynomially-simulated}.

\Cref{fig:forcing-translations} summarizes the different intermediate theories we will consider between $\RCA_0 + \BSig_2$ and $\RT_2^2 + \WKL_0$, and the polynomial simulation between them. It is adapted from \cite[Figure 1]{kolodziejczyk2023ramsey}, with $\BSig_2$ added to every theory, and with a $\forall \Sigma_3^0$ polynomial simulation between
$\RCA_0 + \BSig_2 + \mathbb{I}'$ and $\WKL_0 + \RT_2^2$ instead of a $\forall \Sigma_2^0$ polynomial simulations.

\begin{lemma}\label[lemma]{lem:poly-sim-I1}
 $\RCA_0 + \BSig_2 + (\mathbb{I}1)$ is polynomially simulated by:
 \begin{enumerate}[(a)]
 \item $\RCA_0 + \ISig_2$ with respect to $\forall \Sigma_3^0$ sentences,
 \item $\RCA_0 + \BSig_2 + \neg \ISig_2$ with respect to $\mathcal{L}_2$ sentences.
 \end{enumerate}
\end{lemma}

\begin{lemma}\label[lemma]{lem:polynomially-reflecting}
 The forcing interpretation of $\WKL_0 + \RT_2^2$ in $\RCA_0 + \BSig_2 + \mathbb{I}'$ given by $\mathsf{Cond}, \trianglelefteqslant$, $\Vdash$ of \Cref{def:forcing-interpretation-def} is polynomially $\forall \Sigma_3^0$-reflecting.
\end{lemma}

\begin{lemma}\label[lemma]{lem:polynomial-simulation}
 $\RCA_0 + \mathbb{I}'$ polynomially simulates $\WKL_0 + \RT_2^2$ with respects to $\forall \Sigma_3^0$-sentences.
\end{lemma}

\begin{repmaintheorem}{thm:rt22-polynomially-simulated}
$\RCA_0 + \BSig_2$ polynomially simulates $\WKL_0 + \RT_2^2$ with respect to $\forall \Sigma_3^0$ sentences.
\end{repmaintheorem}

The article is divided into three parts : in \Cref{sec:largeness}, we survey and compare three related notions of largeness, namely, Ketonen and Solvay's $\alpha$-largeness, Le Houérou, Levy Patey and Yokoyama's $\alpha$-largeness$(\theta)$, and our new notion of $\alpha$-largeness${}^*(\theta)$, for $\alpha$ of the form $\bbomega^n \cdot k$. In \Cref{sec:combinatorics}, we develop the framework of $\alpha$-largeness${}^*(\theta)$ and prove polynomial upper bounds for $\RT^2_2$-$\bbomega^n$-largeness${}^*(\theta)$. Last, in \Cref{sec:translation}, we combine this upper bound with the framework of forcing interpretation to prove \Cref{thm:rt22-polynomially-simulated}.

\subsection{Notation}

Most of the proofs are over a weak subsystem of second-order arithmetic. We therefore distinguish the formal set of integers~$\NN$ from the theory from the standard set of integers~$\omega$ from the meta-theory. For instance, if $\M = (M, S)$ is a non-standard model of second-order arithmetic, $\NN^\M = M$ and $\omega^\M = \{0^\M, 1^\M, \dots \} \subseteq M$. Similarly, we distinguish the formal ordinals of the theory and the ordinals from the meta-theory. We therefore write for instance $\bbomega^\bbomega$ for the formal counter-part of~$\omega^\omega$. Indeed, since $\bbomega^{\bbomega} = \sup_{n \in \NN} \bbomega^n$, if $\omega \subsetneq M$, then $\bbomega$ and a fortiori $\bbomega^\bbomega$ are not well-founded from the viewpoint of the meta-theory.

We often identify an integer $k$ with its ordinal $\{0, \dots, k-1\}$.
Given a set $X \subseteq \NN$ and an integer~$n \in \NN$, we write $[X]^n$ for the collection of all subsets of~$X$ of size~$n$.
The set $[X]^n$ is in one-to-one correspondence with the set of all increasing ordered $n$-tuples over~$X$. Therefore, for simplicity of notation, given a coloring $f : [X]^n \to k$, we write $f(x_0, \dots, x_{n-1})$ for $f(\{x_0, \dots, x_{n-1}\})$ assuming that $x_0 < \dots < x_{n-1}$.

The post-theorem introduction gives a high-level proof plan: (1) in \Cref{sec:largeness} it surveys and compares three notions of largeness—Ketonen and Solvay's $\alpha$-largeness, Le Hou\'erou–Levy Patey–Yokoyama's $\alpha$-largeness$(\theta)$, and the new notion $\alpha$-largeness${}^*(\theta)$—for $\alpha$ of the form $\bbomega^n \cdot k$; (2) in \Cref{sec:combinatorics} it develops the framework of $\alpha$-largeness${}^*(\theta)$ and proves polynomial upper bounds for $\RT^2_2$-$\bbomega^n$-largeness${}^*(\theta)$; (3) in \Cref{sec:translation} it "combine[s] this upper bound with the framework of forcing interpretation" to prove \Cref{thm:rt22-polynomially-simulated}.

\label[maintheorem]{thm:rt22-polynomially-simulated}
$\RCA_0 + \BSig_2$ polynomially simulates $\WKL_0 + \RT_2^2$ with respect to $\forall \Sigma_3^0$ sentences.,

Question: Say that a theory \(T_1\) polynomially simulates a theory \(T_0\) with respect to a class \(\Gamma\) of sentences if there exists a polynomial \(Q\) such that for every \(T_0\)-proof \(p\) of any sentence in \(\Gamma\), the translated \(T_1\)-proof \(\hat p\) of the same sentence satisfies \(|\hat p|\le Q(|p|)\). Here a \(\forall\Sigma^0_3\)-sentence means a sentence of the form \(\forall X\,\varphi(X)\), where \(\varphi\) is arithmetical \(\Sigma^0_3\). Which statement holds for the second-order arithmetic theories \(\RCA_0+\BSig_2\) and \(\WKL_0+\RT^2_2\), where \(\BSig_2\) is \(\Sigma^0_2\)-bounding and \(\RT^2_2\) is Ramsey's theorem for pairs and two colors?

\(\RCA_0+\BSig_2\) polynomially simulates \(\WKL_0+\RT^2_2\) with respect to \(\forall\Sigma^0_3\) sentences; equivalently, there is a polynomial \(Q\) such that every \((\WKL_0+\RT^2_2)\)-proof \(p\) of a \(\forall\Sigma^0_3\)-sentence can be translated into an \((\RCA_0+\BSig_2)\)-proof \(\hat p\) of the same sentence with \(|\hat p|\le Q(|p|)\).

Choices:
B. \(\RCA_0+\BSig_2\) polynomially simulates \(\WKL_0+\RT^2_2\) with respect to all \(\mathcal L_2\)-sentences; equivalently, there is a polynomial \(Q\) such that every \((\WKL_0+\RT^2_2)\)-proof \(p\) of an \(\mathcal L_2\)-sentence can be translated into an \((\RCA_0+\BSig_2)\)-proof \(\hat p\) of the same sentence with \(|\hat p|\le Q(|p|)\).
C. \(\RCA_0+\BSig_2\) simulates \(\WKL_0+\RT^2_2\) with respect to \(\forall\Sigma^0_3\) sentences; equivalently, every \((\WKL_0+\RT^2_2)\)-proof of a \(\forall\Sigma^0_3\)-sentence can be translated into an \((\RCA_0+\BSig_2)\)-proof of the same sentence.
D. \(\WKL_0+\RT^2_2\) polynomially simulates \(\RCA_0+\BSig_2\) with respect to \(\forall\Sigma^0_3\) sentences; equivalently, there is a polynomial \(Q\) such that every \((\RCA_0+\BSig_2)\)-proof \(p\) of a \(\forall\Sigma^0_3\)-sentence can be translated into a \((\WKL_0+\RT^2_2)\)-proof \(\hat p\) of the same sentence with \(|\hat p|\le Q(|p|)\).
E. \(\RCA_0+\BSig_2\) simulates \(\WKL_0+\RT^2_2\) with respect to \(\forall\Sigma^0_3\) sentences, but no polynomial bound on the translation length is available; equivalently, for every translation of a \((\WKL_0+\RT^2_2)\)-proof \(p\) of a \(\forall\Sigma^0_3\)-sentence into an \((\RCA_0+\BSig_2)\)-proof \(\hat p\) of the same sentence, one cannot in general ensure \(|\hat p|\le Q(|p|)\) for any polynomial \(Q\).

sentence-class restriction to \forall\Sigma^0_3

polynomial bound on translated proof length