\label{thm:gawron-generalization}
	If $z\in \mathcal S$, then $|z| > \tfrac 14$.

\begin{thm}\label{thm:MO-ineq}
	For all real numbers $t$ and all positive integers $n$,
	\begin{equation}\label{eqn:tao-ineq}
		\abs{\left(1+\frac{it}n\right)^n - 1} \geq \big|e^{it} - 1\big| = 2\sin\frac t2.
	\end{equation}
\end{thm}

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lm$ lie outside the disk $\{\abs{z-2}\leq 1\}\subseteq\C$.
\end{thm}

\begin{thm}\label{thm:gawron-generalization}
	If $z\in \calS$, then $|z| > \tfrac 14$.
\end{thm}

\begin{conj}\label{conj:location-roots}
	All elements of $\calS$ lie in the half-plane $\{\Re w < 1\}$.
\end{conj}

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lm$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lm$ is irreducible in $\Q$.
\end{corr}

The Stern sequence $(s_n)_{n\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and
\begin{equation}\label{eqn:stern-seq}
 s_{2n} = s_n,\quad s_{2n+1} = s_n + s_{n+1}.
\end{equation}
There is a vast body of literature (e.g. \cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\geq 1}$, giving an explicit enumeration of $\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!

\par In \cite{Klavzar2007}, Klav\v{z}ar, Milutinovi\'{c}, and Petr define a polynomial analogue $\stern n\lambda$ of the Stern sequence via $\stern 0\lambda = 0$, $\stern 1\lambda = 1$, and
\begin{equation}\label{eqn:stern-def}
 \stern{2n}\lambda = \lambda \stern n\lambda,\quad \stern{2n+1}\lambda = \stern n\lambda + \stern{n+1}\lambda.
\end{equation}
The first $16$ terms of the sequence $\stern n\lambda$ are given in Table \ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \ref{prop:basic-patterns}.

\begin{table}[ht]
 \input{table-stern-16}
 \caption{\small The first $16$ terms of the sequence $\stern n\lambda$.}
 \label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lambda$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lambda = \lambda \stern n \lambda$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lambda$ only when $n$ is odd. Define
\[
\mathcal S\coloneqq \{z\in\mathbb C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\mathcal S$. For completeness, we record the proof here.

\begin{prop}\label{prop:basic-patterns}
	Let $n$ be a nonnegative integer.

	\vspace{-3pt}

	\begin{enumerate}
		\item $\stern{n}{0} = 0$ when $n$ is even, and $\stern n 0 = 1$ when $n$ is odd.
		\item $\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \eqref{eqn:stern-seq}.
		\item $\stern{n}{2} = n$.
		\item When $n = 2^r$ for a nonnegative integer $r$, $\stern n \lm = \lm^r$.
		\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\stern n\lm = (\lm)_r$.
	\end{enumerate}
\end{prop}

\begin{table}[ht]
	\input{table-stern-16}
	\caption{\small The first $16$ terms of the sequence $\stern n\lm$.}
	\label{table:stern-16}
\end{table}

While there are several papers which analyze the sequence $\stern n\lm$ (see \cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\stern{2n}\lm = \lm \stern n \lm$ for all $n\geq 1$, it suffices to examine the roots of $\stern n\lm$ only when $n$ is odd. Define
\[
\calS\coloneqq \{z\in\C: \stern n z = 0\text{ for some odd }n\geq 1\}.
\]
In \cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\tfrac12$, and $-\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\tfrac14,\tfrac14]$ cannot be in $\calS$. His methods extend easily to show that any complex number $z$ with $|z| \leq \tfrac14$ cannot be in $\calS$. For completeness, we record the proof here.

\begin{proof}
 Let $z$ be any complex number with $|z| \leq \tfrac 14$. Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$. We show more strongly that
 \begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
 \end{equation}
 for all $n\geq 1$.

\begin{thm}[Parabola Theorem, {\cite[Theorem 3.43]{Lorentzen2008}}]\label{thm:parabola}
 For fixed $|\alpha| < \tfrac\pi 2$, let
 \[
 V_\alpha\coloneqq -\tfrac12 + e^{i\alpha}\overline{\mathbb H} = \{w\in\C: \Re(we^{- i\alpha})\geq -\tfrac12\cos\alpha\}
 \]
 and
 \[
 E_\alpha \coloneqq \{a\in\C: |a| - \Re(ae^{-2i\alpha})\leq \tfrac12\cos^2\alpha\}.
 \]
 Then $E_\alpha$ is the element set for continued fractions $\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\alpha$.
\end{thm}
The region $V_\alpha$ is a half-plane whose boundary is a line intersecting the real axis at $z = -\tfrac12$. In particular, because $|\alpha| < \tfrac\pi 2$, $V_\alpha$ contains the half-line $[-\tfrac12,\infty)$. The boundary of $E_\alpha$ is a parabola with focus at the origin and vertex at $-\tfrac14 e^{2i\alpha}\cos^2\alpha$. This parabola intersects the real axis at $z = -\tfrac14$.

We now tie the continued fraction theory back to Stern polynomials.
Recall that Corollary \ref{thm:stern-cont-frac} expresses the ratio
of two Stern polynomials as a continued fraction with elements in $\C$.
This form on its own is difficult to work with. We instead note (e.g. \cite[Corollary 2.15]{Lorentzen2008}) that any continued fraction $\boldsymbol{K}(a_n|b_n)$ with $b_j\neq 0$ is equivalent to the continued fraction $\boldsymbol{K}(c_n|1)$, where $c_j = \tfrac{a_j}{b_jb_{j+1}}$. As an explicit example,
\begin{equation}\label{eqn:cfrac-example}
 \frac{\stern{[[2,3,5]]}\lm}{\stern{[[3,5]]}\lm} = (\lm)_2 + \cfrac{\lm^2}{(\lm)_3 + \cfrac{\lm^3}{(\lm)_5}}
 = (\lm)_2 \left[1 + \cfrac{\dfrac{\lm^2}{(\lm)_2(\lm)_3}}{1 + \cfrac{\lm^3}{(\lm)_3(\lm)_5}}\right].
\end{equation}
\begin{remark}
 While continued fractions of the form $\boldsymbol{K}(1|d_n)$ are more common, the coefficients $d_n$ are significantly messier than the coefficients $c_n$ (see \cite[Corollary 2.15]{Lorentzen2008}). We opt to use the less-common form $\boldsymbol{K}(c_n|1)$ to simplify our analysis.
\end{remark}
Equation \eqref{eqn:cfrac-example} suggests that the ratios $\tfrac{z^a}{(z)_a(z)_b}$, where $a$ and $b$ are any positive integers, may play some importance.
For this reason, we define $z_{a,b}\coloneqq \tfrac{z^a}{(z)_a(z)_b}$ and
\[
\calA_z\coloneqq \left\{z_{a,b}:(a,b)\in\N^2\right\}.
\]
Suppose $\calA_z\subseteq E_\alpha$ for some angle $\alpha$ with $|\alpha| < \tfrac{\pi}2$.
Then $V_\alpha$ is a value set for the continued fraction \eqref{eqn:stern-cont-frac} at $\lm = z$. In particular, $\infty$ is not a possible value for this fraction, so $\stern n z \neq 0$ for any $z$.

\begin{itemize}
 \item First suppose $x\in[0,\pi]$. Because both $\sin x$ and $x - \tfrac{x^2}{\pi}$ are symmetric about the axis $x = \tfrac\pi 2$, it suffices to prove the inequality for $x\in[0,\tfrac\pi 2]$. Within this smaller interval, cosine is concave down with $\cos 0 = 1$ and $\cos\tfrac \pi 2 = 0$, so
 \begin{equation}\label{eqn:cos-ineq}
 \cos x \geq 1 - \frac{2x}{\pi}.
 \end{equation}
 Integrating both sides of \eqref{eqn:cos-ineq} yields \eqref{ineq-sinc} in this case.
 \item Now suppose $x\notin[0,\pi]$. By symmetry, it suffices to prove the inequality for $x\geq \pi$. In this interval, $1 - \tfrac{2x}{\pi} \leq -1 \leq \cos x$. It follows that \eqref{eqn:cos-ineq}, and thus \eqref{ineq-sinc}, holds as well.
 \end{itemize}
\end{proof}

\begin{thm}\label{thm:min-geom-series}
 Let $n$ be a positive integer, and suppose $z\in\calB$. Then $|(z)_n| \geq \min(n,\tfrac{11}2)$.
\end{thm}

Now let
 \[
 M\coloneqq 2\pi\left(1 - \frac{5.5}{n}\right) = \pi\left(2 - \frac{11}n\right).
 \]
 There are two cases to consider. First, suppose $t\leq M$. Then
 \[
 n\left(1 - \frac{t}{2\pi}\right) \geq n\left(1 - \frac{M}{2\pi}\right) = \frac{11}2.
 \]
 Now suppose $t\geq M$. Because $n\geq 15$, the coefficient $\tfrac{1}{2n} - \tfrac{11}{8n^2}$ is positive, so
 \begin{equation}\label{eqn:asymp-w-tn}
 t + t^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right) \geq \pi\left(2 - \frac{11}n\right) + \pi^3\left(2 - \frac{11}n\right)^3\left(\frac{1}{2n} - \frac{11}{8n^2}\right).
 \end{equation}
 To estimate \eqref{eqn:asymp-w-tn}, let $x = \tfrac 1n$, so it suffices to analyze the polynomial
 \[
 P(x) := \pi(2-11x) + \pi^3(2-11x)^3(\tfrac 12x - \tfrac{11}{8}x^2).
 \]
 The polynomial $P$ has exactly one critical point in the interval $[0,\tfrac 1{15}]$, occurring at $x = x_0 \approx 0.024$. In particular, $P$ is increasing on $[0,x_0]$ and decreasing on $[x_0,\tfrac1{15}]$. Compute
 \[
 P(0) = 2\pi\quad\text{and}\quad P\left(\frac 1{15}\right) = \frac{19 \pi }{15}+\frac{336091 \pi ^3}{6075000} \approx 5.695 > \frac{11}2.
 \]
 It follows that $P(x) > \tfrac{11}2$ for all $x\in[0,\tfrac{1}{15}]$, and thus $|w_{n,t}^n - 1| \geq M|w_{n,t}-1|$ for all $t\in[0,2\pi]$ and $n\geq 15$.
\end{proof}

\begin{prop}\label{prop:silli}
 Let $z\in\calB^+$. Then $0\leq \arg z \leq \tfrac\pi 6$ and
 \begin{equation}\label{eqn:re-ineq}
 |\Re z^{-4}|\leq |\Re z^{-2}|.
 \end{equation}
\end{prop}

\begin{corr}[{\cite[Theorem 1]{Schinzel2014}}]\label{thm:stern-cont-frac}
	Let $a_1,\ldots, a_t$ be positive integers, where $t\geq 2$.  Then
	\begin{equation}\label{eqn:stern-cont-frac}
		\frac{\stern{[[a_1,\ldots, a_t]]}{\lm}}{\stern{[[a_2,\ldots, a_t]]}{\lm}}
		=
		(\lm)_{a_1} + \cfrac{\lm^{a_1}}{(\lm)_{a_2} + \cfrac{\lm^{a_2}}{\cdots + \cfrac{\cdots}{(\lm)_{a_{t-1}} + \cfrac{\lm^{a_{t-1}}}{(\lm)_{a_t}}}}}\,.
	\end{equation}
\end{corr}

\begin{proof}
	Let $z$ be any complex number with $|z| \leq \tfrac 14$.  Let $b_n\coloneqq |\stern n z|$ for each positive integer $n$.  We show more strongly that
	\begin{equation}\label{eqn:stronger-statement}
		b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
	\end{equation}
	for all $n\geq 1$.

	\par The proof of \eqref{eqn:stronger-statement} proceeds by induction on $n$.  The base case, $n = 1$, follows because
	\[
	b_3 = |z + 1| \geq \tfrac 34 > \tfrac 12
	= \tfrac 12\max\{1, |z|\} = \tfrac 12\max\{b_1,b_2\}.
	\]

	\par There are two cases to consider.  First assume $n=2k$ is even.  Then
	\begin{align*}
		b_{4k+1} &= |\stern{4k+1}z| = |\stern{2k+1} z + \stern{2k} z|\\
		&= |z \stern kz + \stern{2k+1} z| \geq b_{2k+1} - \tfrac14 b_k\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_k$, and thus $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  In this case, our inequality is proved.

	\par Now assume $n = 2k + 1$ is odd.  Then
	\begin{align*}
		b_{4k+3} &= |\stern{4k+3}z| = |\stern{2k+1} z + \stern{2k+2} z|\\
		&= |\stern{2k+1} z + z\stern{k+1} z| \geq b_{2k+1} - \frac 14 b_{k+1}\\
		&\geq b_{2k+1} - \tfrac12 b_{2k+1} = \tfrac12 b_{2k+1}.
	\end{align*}
	Moreover, $b_{2k+1} \geq \frac12 b_{k+1}$, and so in this case we also have $\max\{b_{2k},b_{2k+1}\} = b_{2k+1} > 0$.  We have exhausted both cases, completing the proof of Theorem \ref{thm:gawron-generalization}.
\end{proof}

In \cite{Dilcher2017}, Dilcher et. al. focus more specifically on the complex roots of $\stern n\lambda$.  Their paper makes the following conjecture.

\begin{conj}\label{conj:location-roots}
	All elements of $\mathcal S$ lie in the half-plane $\{\operatorname{Re} w < 1\}$.
\end{conj}

By generalizing the Enestrom-Kakeya theorem, they prove Conjecture \ref{conj:location-roots} for several classes of positive integers $n$ taking the form $2^n\pm k$, where $k$ is fixed and $2^n \geq k$.  These are the only two papers the author could find which discuss the complex zeros of $\stern n\lambda$.

\par Figure \ref{fig:stern-roots} shows a snapshot of $\mathcal S$.  One striking feature of this figure is the contrasting behavior of these roots within the half-planes $\{\operatorname{Re} w\geq 0\}$ and $\{\operatorname{Re} w < 0\}$.  These differences present difficulties in fully characterizing the geometry of $\mathcal S$.

\begin{figure}[ht]
	\centering
	\captionsetup{width=0.75\linewidth}
	\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}
	\caption{\small The zeros of $\stern n\lambda$ in the range $\{a+bi:-4\leq a\leq 1, |b|\leq 3\}$, where $1\leq n<2^{21}$ is odd.}
	\label{fig:stern-roots}
\end{figure}

In this paper, we partially resolve Conjecture \ref{conj:location-roots} by establishing the following result.

\begin{thm}\label{thm:roots-disk}
	All roots of $\stern n\lambda$ lie outside the disk $\{\left|z-2\right|\leq 1\}\subseteq\mathbb C$.
\end{thm}

To prove Theorem \ref{thm:roots-disk}, we use a continued fraction representation for a ratio of Stern polynomials (Theorem \ref{thm:stern-cont-frac}) independently discovered by Reznick \cite{Reznick2008} and Schinzel \cite{Schinzel2014}.  This allows us to use the Parabola Theorem (Theorem \ref{thm:parabola}) to show that, for certain values of $z\in\mathbb C$, the denominators of these continued fractions can never be zero.  Along the way, we establish inequalities in $\mathbb C$ relating to the sums $1 + z + \cdots + z^{n-1}$ which may be of independent interest.  The most notable of these inequalities is Theorems \ref{thm:MO-ineq}, which proves a lower bound for this geometric series whenever $\operatorname{Re} z \geq 1$.

As a corollary, we obtain the following surprising fact, resolving a conjecture of Ulas and Ulas (\cite{Ulas2011}).

\begin{corr}\label{thm:prime-poly-irred}
	For each prime number $p$, the Stern polynomial $\stern p \lambda$ is irreducible in $\mathbb Q$.
\end{corr}
Ulas and Ulas had verified this conjecture computationally for the first million primes $p$.  Additionally, Schinzel in \cite{Schinzel2011} proved Corollary \ref{thm:prime-poly-irred} for all primes $p < 2017$ by using finite differences to bound the leading coefficient of any proper divisor of $\stern n\lambda$.  However, these previous attempts to prove the conjecture were algebraic in nature, whereas our proof depends on the analytic properties of $\mathcal S$.

\paragraph{Acknowledgments.} This paper is adapted from the author's PhD dissertation \cite{Altizio2025}.  The author thanks his advisor, Dr. Bruce Reznick, for helpful correspondence.

Let $z$ be any complex number with $|z|\leq \tfrac14$ and set $b_n\coloneqq |\stern n z|$. The proof shows the stronger claim
\begin{equation}\label{eqn:stronger-statement}
 b_{2n+1} > \tfrac12\max\{b_n,b_{n+1}\} > 0
\end{equation}
for all $n\ge1$, and proceeds by induction on $n$.

Base case $n=1$:
\[
 b_3=|z+1|\ge \tfrac34>\tfrac12=\tfrac12\max\{1,|z|\}=\tfrac12\max\{b_1,b_2\}.
\]

Inductive step splits into two cases.

(i) If $n=2k$ is even, then
\[
 b_{4k+1}=|\stern{4k+1}z|=|\stern{2k+1}z+\stern{2k}z|=|z\,\stern k z+\stern{2k+1}z|
 \ge b_{2k+1}-\tfrac14 b_k\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_k$, hence $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$, giving \eqref{eqn:stronger-statement}.

(ii) If $n=2k+1$ is odd, then
\[
 b_{4k+3}=|\stern{4k+3}z|=|\stern{2k+1}z+\stern{2k+2}z|=|\stern{2k+1}z+z\,\stern{k+1}z|
 \ge b_{2k+1}-\tfrac14 b_{k+1}\ge b_{2k+1}-\tfrac12 b_{2k+1}=\tfrac12 b_{2k+1}.
\]
Moreover, $b_{2k+1}\ge \tfrac12 b_{k+1}$, so again $\max\{b_{2k},b_{2k+1}\}=b_{2k+1}>0$. These two cases exhaust the induction, completing the proof of Theorem~\ref{thm:gawron-generalization}.

Question:
Let the Stern polynomials \(\operatorname{St}_n(\lambda)\) be defined by \(\operatorname{St}_0(\lambda)=0\), \(\operatorname{St}_1(\lambda)=1\), and for all integers \(n\ge 0\),
\[
\operatorname{St}_{2n}(\lambda)=\lambda\operatorname{St}_n(\lambda),\qquad \operatorname{St}_{2n+1}(\lambda)=\operatorname{St}_n(\lambda)+\operatorname{St}_{n+1}(\lambda).
\]
Define
\[
\mathcal S=\{z\in\mathbb C: \operatorname{St}_n(z)=0\text{ for some odd }n\ge 1\}.
\]
If \(z\in \mathcal S\), which conclusion about \(z\) holds?

One must have \(|z|>\tfrac14\).

Choices:
B. One must have \(|z|\ge \tfrac14\).
C. One must have \(z\neq 0\).
D. One must have \(\Re(z)<-\tfrac14\).
E. Every zero of a Stern polynomial satisfies \(|z|>\tfrac14\), whether the index \(n\) is odd or even.

numerical lower bound on |z| weakened to mere nonvanishing

modulus condition replaced by a real-part restriction

The stem gives only the definition of the Stern polynomials and the set \(\mathcal S\); it does not explicitly state the target conclusion. There is no direct answer leakage beyond the general framing of asking for a consequence about zeros.

This is essentially a direct theorem-recall item: the correct option is the main conclusion one would prove about \(z\in\mathcal S\), namely \(|z|>\tfrac14\). The question mostly asks the student to recognize the theorem rather than infer a new consequence from competing premises.

There is some reasoning pressure because the options include a weaker true statement (\(z\neq 0\)), a boundary variant (\(|z|\ge \tfrac14\)), and a quantifier overreach. Still, if the underlying result is known, the answer is immediate, so the item tests recognition more than substantial generation.

The distractors are well-designed: one weakens strictness, one gives a weaker true claim, one changes the type of restriction, and one improperly extends from odd indices to all indices. These are distinct and reflect realistic mathematical failure modes.

A reasonably well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it is highly tautological and only moderately tests genuine generative reasoning.


\label{thm0}

		Let $M$ be a closed orientable $n$-manifold with an open book decomposition $\textrm{OB}(P^{n-1},\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\partial P$ and let $\textrm{OB}(V_B,\phi_B)$ be an open book decomposition of $B$. Then, $\textrm{OB}(V_B, \phi_B^k)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$ for all $k \in \mathbb{Z}$.

\begin{corollary} \label{cor2}

		Let $Q$ be a closed simply connected spin $5$-manifold. Then, every open book decomposition of $Q$ spun embeds in $\s^7$ with a fixed page bounded by $Q$.

	\end{corollary}

\begin{corollary}\label{cor3}

		Let $M^3$ be a closed orientable $3$-manifold. Then, every open book decomposition of $M^3$ spun embeds in $\s^5$ with a fixed page bounded by $M$.

	\end{corollary}

\begin{enumerate}
			\item  $\Gamma_k$ is supported on $\mathcal{C}(B)$ and $\Gamma_k$ is isotopic to the identity map, relative to $P \setminus \mathcal{C}(B)$.

			\item $\Gamma_k \circ f = f \circ \phi_B^k$.
		\end{enumerate}

		\noindent Since $\rho|_{\mathcal{C}(B)} = \id_P$, $\ob(V_B, \phi^k_B)$ spun embeds in $\ob(P,\rho)$.

	\end{proof}

	\section{The page trick in a more general setup} \label{pagetrickap}

	Let $M^n$ be a component of the binding of on open book $\ob(\Sigma,\phi)$. Let $\ob(P_1^{n-1}, \rho_1)$ and $\ob(P_2^{n-1}, \rho_2)$ be two open book decompositions of a closed oriented manifold $M^n$. Let $g_i$ denote the embedding of $P_i$ in $M$ as a fixed page of the open books for $i = 1,2$. Let $V$ be a compact bounded submanifold of $M$ such that $g_1(P_1) \cup g_2(P_2)$ is a submanifold of $V$.

	\begin{lemma} \label{genpagetrick}
	$\ob(V, \eta)$ spun embeds in $\ob(\Sigma, \phi)$ for any $\eta \in \langle \rho_1, \rho_2\rangle \subset \Diff_\partial(V)$.
	\end{lemma}

	\begin{exmp}\label{expg1}

	The $5$-sphere admits an open book decomposition with page the disk cotangent bundle of $\s^2$ and monodromy a Dehn-seidel twist : $\ob(DT^*\s^2, \tau_2)$ (see \cite{Saha}). Another open book on $\s^5$ has page is given by $\ob(\s^1 \times \D^3 \#_b \s^3 \times [0,1], \sigma)$, where $\sigma$ is the \emph{push} map (see \cite{hsueh1}, \cite{hsueh2}). Let $g_1$ and $g_2$ be the embeddings of $P_1 = DT^*\s^2$ and $P_2 = \s^1 \times \D^3 \#_b \s^3 \times [0,1]$, respectively, in $\s^5$. Then, $g_1(P_1) \#_b g_2(P_2)$ gives a hypersurface in $\s^5 = \s^5 \# \s^5$. By Lemma \ref{genpagetrick}, for any diffeomorphism $\eta$ of $g_1(P_1) \#_b g_2(P_2)$ generated by $\tau_2$ and $\sigma$, $\ob(g_1(P_1) \#_b g_2(P_2), \eta)$ spun embeds in $\ob(\D^6,\id)$.  \noindent Hsueh \cite{hsueh2} has given a large family of new open book decompositions on $\s^n$. Using his open books, one can generate many more examples as the one above.

	\end{exmp}

	\begin{remark}
		It follows from Theorem \ref{thm0} that $M_k = \ob(\s^1 \times \D^2 \#_b \s^2 \times [0,1], \sigma^k)$ spun embeds in $\ob(\D^5,\id) = S^6$. Since $H_1(M_k) = \Z_k$, by a result of Cochran (see Theorem $6.2$ in \cite{coch}), $M_k$ also embeds in $\s^5$ for all $k \geq 0$. It would be interesting to know whether $M_k$ spun embeds in $\s^5$ for all $k \geq 2$.
	\end{remark}

	\begin{exmp}\label{expg2}

	In example \ref{expg1}, we obtained the page as boundary connected sum of pages of two open books. Suppose we have two pages $P^n_1$ and $P^n_2$ such that there exist relative embeddings $e_1 : (\D^k \times \D^{n-k}, \partial \D^k \times \D^{n-k})  \rightarrow (P_1, \partial P_1)$ and $e_2 : (\D^{n-k}\times \D^k, \partial \D^{n-k} \times \D^k) \rightarrow (P_2, \partial P_2)$. Then one can define the plumbing of $P_1$ and $P_2$ by gluing the images of $e_1$ and $e_2$ via the map $e_1(\vec{x},\vec{y}) \mapsto e_2((-1)^k \vec{y}, \vec{x})$. Ozbagci and Popescu-Pampu has defined a generalized notion of plumbing where two manifolds such as $P_1$ and $P_2$ can be \emph{summed} using a common \emph{patch} (see section $7$ in \cite{OP}).

	\end{exmp}

	\begin{figure}[htbp]

		\centering
		\def\svgwidth{10cm}
		\input{fig02.pdf_tex}
		\caption{The purple and green lines represent $g_1(P_1)$ and $g_2(P_2)$, respectively,  in $M \times [-2,1]$ and $s$ is a coordinate on the second factor of $\Sigma \times [0,1]$. We first isotope $g_1(P_1)$ to induce $\rho_1$ at $s = \frac{1}{3}$, and then isotope $g_2(P_2)$ to induce $\rho_2$ at $s =\frac{2}{3}$. This induces the monodromy $\rho_1 \circ \rho_2$ on $V$.}
		\label{gptfig}

	\end{figure}

	\begin{proof}[Proof of Lemma \ref{genpagetrick}]  We identify a collar neighborhood of $M \subset \Sigma$ with $M \times [-2,2]$ such that $\partial \Sigma = M \times \{2\}$. We take a Seifert type proper embedding of $V$ in $M \times [-2,-1]$. Suppose, we want to induce the map $\rho_1$ on $V$ first. For that we simply isotope the submanifold $g_1(P_1) \times \{-1\}$, relative to its boundary, such that after the isotopy $g_1(P_1) \times \{-1\}$ becomes a Seifert type embedding $f_1$ of $P_1$ in $M \times [-1,0]$. We then apply the page trick in the neighborhood $M \times [-1,1]$ to induce the map $\rho_1$, and isotope $f_1(P_1)$ back to $g_1(P_1) \times \{-1\}$. Similarly, one can induce the map $\rho_2$ on the properly embedded $V$ in $M \times [-2,2]$. See Figure \ref{gptfig}. Let $(\Sigma \times [0,1], (x,\theta))$ be a coordinate on the mapping cylinder of $\ob(\Sigma, \phi)$. Say, $\eta$ has a presentation as a word generated by $\rho_1$ and $\rho_2$ with $l$ many letters. We divide the mapping cylinder in $l$ equal part $\Sigma \times [\frac{i}{l}, \frac{i+1}{l}]$ ($0 \leq i \leq l-1$), and apply an isotopy of $\Sigma$ on each part to induce each of the letters. Thus, $\ob(V,\eta)$ spun embeds in $\ob(\Sigma, \phi)$.

	\end{proof}

	\noindent Although Lemma \ref{genpagetrick} is stated for two open books, its proof shows that a similar statement holds true for any number of open books. The page trick can be used for constructing $5$-dimensional open books with connected binding where all $3$-dimensional open books with connected binding admit spun embedding (see Theorem $4.4$ in \cite{ls1}). Examples of such $5$-dimensional open books were first constructed by Etnyre and Lekili \cite{EL} using mapping torus of a Lefschetz fibration.

	\section{The page trick for Morse open books} \label{morseob}

		A \emph{Morse open book} is similar to an open book $(M^n, B^{n-1},\pi)$, except that the fibration map $\pi : M \setminus B \rightarrow \s^1$ may have singular fibers, with singularities in the interiror of a fiber.

	   \noindent  Let $K^{n-2}$ be a closed oriented submanifold of a closed oriented manifold $L^n$ with trivial normal bundle and let $K$ be nullhomologous in $L$. Then, there exists a hypersurface $V^{n-1} \subset L$ such that $\partial V = K$. Given such a \emph{Seifert surface} $V$, there exists a Morse open book on $L$ with a regular page $V$ (see remark $9.6$ in \cite{OP}). In particular, $L \setminus V$ can be seen as a relative cobordism $W^n$ from $V$ to itself. Let $\psi$ be the monodromy map of the regular page $V$. Let $h : W^n \rightarrow [0,1]$ be a Morse function with $h^{-1}(0) = h^{-1}(1) = V$ and $h$ has critical points only in the interior of $W$. According to Milnor's terminology, $h$ is a Morse function on the \emph{triad} $(W, h^{-1}(0), h^{-1}(1))$. The monodromy $\psi$ is then a diffeomorphism (relative to boundary) between $h^{-1}(0)$ and $h^{-1}(1)$. In this case, the analogue of the mapping torus of an open book is the quotient space $\mathcal{M}(W,h,\psi) = \frac{W}{x \sim \psi(x)}$. Thus, the triple $(W,h,\psi)$ characterizes a Morse open book structure on $L$.

	   \begin{definition}[spun embedding for Morse open books]

	   	Let $L_1^n$ and $L_2^{n+k}$ be two closed oriented manifolds. Let $(W_1,h_1, \psi_1)$ and $(W_2,h_2,\psi_2)$ be Morse open book decompositions of $L_1$ and $L_2$, respectively. We say $(W_1, h_1, \psi_1)$ spun embeds in $(W_2,h_2,\psi_2)$ if there exists an embedding $\iota$ of $L_1$ in $L_2$ such that the followings hold.

	   	\begin{enumerate}
	   		\item $h_2 \circ \iota$ restricts to a Morse function on the triad $(W_1, h_1^{-1}(0), h_1^{-1}(1))$.

	   		\item  The embedding $\iota$ takes a regular page of $h_1$ to a regular page of $h_2$ and $\iota \circ \psi_1 | _{h_1^{-1}(1)}= \psi_2 \circ \iota|_{h_1^{-1}(1)}$.

	   	\end{enumerate}

\begin{proposition} \label{morsepagetrick}

		Let $(W, h, \phi)$ be a Morse open book decomposition of a closed oriented manifold $N$. Then, there exists an embedding $F : W \rightarrow N \times [-1,1] \times [0,1]$ with the following property.

		\begin{enumerate}

			\item The projection map $N \times [-1,1] \times [0,1] \rightarrow [0,1]$ restricts to a Morse function on the triad $(F(W), F(h^{-1}(0)), F(h^{-1}(1)))$, and

			\item $F \circ \phi|_{h^{-1}(1)} = F|_{h^{-1}(0)}$.

		\end{enumerate}

		\end{proposition}

\label{genhopftrick}

	The Hopf annulus trick has a straightforward generalization for any open book decomposition. Let $N^n = \ob(V^{n-1},\phi)$. Consider the Seifert type embedding $h$ of $V$ in $N\

\begin{exmp} \label{brancexmp}

			Let $K^{n-2}$ be a closed oriented submanifold of a closed oriented manifold $L^n$ with trivial normal bundle and let $K$ be nullhomologous in $L$. Thus, there exists a Morse open book $(W, h, \phi)$ on $L$, such that the closure of $L \setminus K$ is the relative cobordism $W$ between a regular page $V$ to itself and the monodromy or the return map is $\phi$. We partition the trivial mapping cylinder $(L \times [-1,1]) \times [0,k]$ in $k$ parts : $L \times [-1,1] \times [i,i+1]$ ($0 \leq i \leq k-1$). By Proposition \ref{morsepagetrick}, we can embed a copy of $W$ in $L \times [-1,1] \times [i,i+1]$ via an embedding $F_i$ such that the projection map $L \times [-1,1] \times [i,i+1] \rightarrow [i,i+1]$ restricts to a Morse function on the triad $(F_i(W), F_i(h^{-1}(0)), F_i(h^{-1}(1)))$, and $F_i \circ \phi|_{h^{-1}(1)} = F_i|_{h^{-1}(0)}$. By concatenating $F_0, F_1,\dots, F_{k-1}$, we get a spun embedding of a Morse open book $(W_k, h_k, \phi^k)$ in $\ob(L \times [-1,1],\id) = L \times \s^2$, where $W_k$ is a relative cobordism obtained by concatenating $k$ copies of $W$ and $h_k$ is the corresponding Morse function given by $h \ast h \ast \cdots \ast h$ ($k$ times). By construction, the total space $\widetilde{L}$ of the Morse open book $(W_k, h_k, \phi^k)$ is a $k$-fold cyclic branched cover of $L$ branched along $K$.

		\end{exmp}

An open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\mathcal{MT}(V^{m-1}, h) \cup_{id} \partial V^{m-1} \times D^2$. Here, $V^{m-1}$ is called the \emph{page} and $\partial V$ is called the \emph{binding} of the open book. The map $h$, called \emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\partial V$, and $\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\textrm{OB}(V,h)$.

\noindent An embedding $f$ of $\textrm{OB}(V_1,h_1)$ in $\textrm{OB}(V_2,h_2)$ is called a \emph{spun embedding} (or an \emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \circ f = f \circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \emph{spun embedding} (or \emph{open book embedding}) has attracted much attention (eg. \cite{EF}, \cite{EL}, \cite{pps}, \cite{GPS}, \cite{Saha} ,\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \cite{ls1}. Etnyre and Lekili \cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \cite{pps}, based on the observation that there is a proper embedding of a annulus $\mathcal{A}$ in $\mathbb{D}^4$ such that, the Dehn twist along the core of $\mathcal{A}$ is induced by an isotopy of $\mathbb{D}^4$, relative to the boundary. This construction, called the \emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \cite{GPS}. In this note, we observe a generalization of this construction and give some applications.

We prove that if a closed manifold $B$ is a connected component of the binding of an open book decomposition of a manifold $M$, then every open book decomposition of $B$ spun embeds in $M$. As an application, we prove that every open book decomposition of a simply connected spin $5$-manifold spun embeds in $\s^7$ and every $3$-dimensional open book spun embeds in $\s^5$. We also define a notion of spun embedding for Morse open books.

An open book decomposition of a manifold $M^m$ is a pair $(V^{m-1},h)$, such that $M^m$ is diffeomorphic to the quotient space $\mathcal{MT}(V^{m-1}, h) \cup_{id} \partial V^{m-1} \times D^2$. Here, $V^{m-1}$ is called the \emph{page} and $\partial V$ is called the \emph{binding} of the open book. The map $h$, called \emph{monodromy}, is a diffeomorphism of $V^{m-1}$ that restricts to identity near the boundary $\partial V$, and $\mathcal{MT}(V^{m-1}, h)$ denotes the mapping torus of $h$. We denote the total space of such an open book by $\textrm{OB}(V,h)$.

\noindent An embedding $f$ of $\textrm{OB}(V_1,h_1)$ in $\textrm{OB}(V_2,h_2)$ is called a \emph{spun embedding} (or an \emph{open book embedding}), if $f$ takes the page $V_1$ to the page $V_2$ and $f$ is compatible with the monodromies : $h_2 \circ f = f \circ h_1$, up to isotopy. We say $M^n$ spun embeds in $N^{n+k}$ if there exists an open book decomposition of $M$ that spun embeds in an open book decomposition of $N$. In recent times, the study of codimension $2$ \emph{spun embedding} (or \emph{open book embedding}) has attracted much attention (eg. \cite{EF}, \cite{EL}, \cite{pps}, \cite{GPS}, \cite{Saha} ,\cite{ls1}). The notion of spun embedding of an open book decomposition is strictly stronger than ordinary smooth embedding. Obstructions to spun embedding an open book in a given open book has been discussed in \cite{ls1}. Etnyre and Lekili \cite{EL} gave the first constructions of codimension $2$ spun embeddings of $3$-manifolds by embedding pages as fibers in a Lefschetz fibration. Later, an alternate construction of spun embedding was given by Pancholi, Pandit and the fourth author \cite{pps}, based on the observation that there is a proper embedding of a annulus $\mathcal{A}$ in $\D^4$ such that, the Dehn twist along the core of $\mathcal{A}$ is induced by an isotopy of $\D^4$, relative to the boundary. This construction, called the \emph{annulus trick}, was later adapted to the setting of non-orientable $3$-manifolds by Ghanawat, Pandit and Selvakumar \cite{GPS}. In this note, we observe a generalization of this construction and give some applications.

Let $M$ be a closed orientable $n$-manifold with an open book decomposition $\ob(P^{n-1},\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\partial P$ and let $\ob(V_B,\phi_B)$ be an open book decomposition of $B$. Then, $\ob(V_B, \phi_B^k)$ spun embeds in $\ob(P^{n-1},\rho)$ for all $k \in \Z$.

\noindent Here, $\phi_B^k$ denotes the $k$-times composition of $\phi_B$. An immediate corollary of Theorem \ref{thm0} is the following.

\noindent Let $Q^5$ be a simply connected spin manifold. Kwon and Van Koert (Corollary $3.9$ in \cite{KoVan}) have shown that every simply connected spin $5$-manifold is a connected sum of Brieskorn manifolds. Thus, $M = B_1 \# B_2 \# \cdots \# B_l$ for some $l \in \Z_{>0}$, where the $B_i$s are $5$-dimensional Brieskorn manifolds. Let $\ob(V_i, \phi_i)$ be an open book decomposition of $\s^7$ with binding $\partial V_i = B_i$. Then, $\ob(V_1 \#_b V_2 \#_b \cdots \#_b V_l, \phi_1 \circ \phi_2 \circ \phi_l) = \ob(V_1,\phi_1) \# \ob(V_2,\phi_2) \# \cdots \# \ob(V_l,\phi_l)$ (see section $2.4$ in \cite{DGK}) is an open book decomposition of $\s^7$ with binding $Q^5$. Thus, Theorem \ref{thm0} implies the following.

\section{preliminaries} \label{prelim}
 \subsection{Open books and embeddings} An open book decomposition of a closed $(2n+1)$-manifold $M$ consists of a codimension $2$ closed submanifold $B$ and a fibration map $\pi : M \setminus B \rightarrow \s^1$, such that in a tubular neighborhood of $B \subset M$, the restriction map $\pi : B \times (\D^2 \setminus \{0\}) \rightarrow \s^1$ is given by $(b,r,\theta) \mapsto \theta$. The fibration $\pi$ determines a unique fiber manifold $N^{2n}$ whose boundary is $B$. The closure $\bar{N}$ is called the \emph{page} and $B$ is called the \emph{binding}. The monodromy of the fibration map $\pi$ determines a diffeomorphism $\phi$ of $\bar{N}$ such that $\phi$ is identity near the boundary $\partial \bar{N}$. In particular, $M = \mathcal{MT}(\bar{N}, \phi) \cup _{id, \partial} \partial \bar{N} \times \D^2$. We denote such an open book decomposition of $M$ by $\textrm{OB}(\bar{N},\phi)$. The map $\phi$ is called the \emph{monodromy} of the open book. This description of an open book decomposition in terms of its page and monodromy is known as an \emph{abstract open book}. However, these two notions are equivalent and we will always describe an open book decomposition of a manifold as an abstarct open book. Two open book decompositions with the same page are \emph{equivalent} if their monodromies are isotopic, relative to a collar neighborhood of the binding in a page.

\begin{proof}[Proof of Theorem \ref{thm0}] Recall that $M^n = \ob(P^{n-1}, \rho)$, where $\rho$ restricts to the identity map on a collar neighborhood $\mathcal{C}(B)$ of $B = \partial P$ in $P$. Let us identify this collar neighborhood with $B \times [-1,1]$ such that $B \times \{-1\} = \partial P$. Consider an open book decomposition $\ob(V_B,\phi_B)$ of $B$ and let $f$ be the Seifert type embedding of $V_B$ in $B \times [-1,1]$ such that $f(\partial V_B) \subset B \times \{-1\}$. This gives a proper embedding of $V_B$ in $P$. Given an integer $k$, by Proposition \ref{pagetrick}, there exists a diffeomorphism $\Gamma_k$ of $P$ with the following properties.

\begin{theorem} \label{thm0}

		Let $M$ be a closed orientable $n$-manifold with an open book decomposition $\ob(P^{n-1},\rho)$. Let $B^{n-2}$ be a connceted component of the binding $\partial P$ and let $\ob(V_B,\phi_B)$ be an open book decomposition of $B$. Then, $\ob(V_B, \phi_B^k)$ spun embeds in $\ob(P^{n-1},\rho)$ for all $k \in \Z$.

	\end{theorem}

\noindent Here, $\phi_B^k$ denotes the $k$-times composition of $\phi_B$. An immediate corollary of Theorem \ref{thm0} is the following.

	\begin{corollary} \label{cor1}
		Every open book decomposition of $\mathbb{S}^{n-2}$ spun embeds in $\mathbb{S}^n = \textrm{OB}(\mathbb{D}^{n-1}, \textrm{id})$ for $n \geq 4$.
	\end{corollary}

	\noindent More generally, one can think of any codimension $2$ fibered submanifold of $\mathbb{S}^m$. In particular, every $(2n-1)$-dimensional Brieskorn manifold appears as the binding of an open book decomposition of $\mathbb{S}^{2n+1}$.

	 \noindent Let $Q^5$ be a simply connected spin manifold. Kwon and Van Koert (Corollary $3.9$ in \cite{KoVan}) have shown that every simply connected spin $5$-manifold is a connected sum of Brieskorn manifolds. Thus, $M = B_1 \# B_2 \# \cdots \# B_l$ for some $l \in \Z_{>0}$, where the $B_i$s are $5$-dimensional Brieskorn manifolds. Let $\textrm{OB}(V_i, \phi_i)$ be an open book decomposition of $\mathbb{S}^7$ with binding $\partial V_i = B_i$. Then, $\textrm{OB}(V_1 \#_b V_2 \#_b \cdots \#_b V_l, \phi_1 \circ \phi_2 \circ \phi_l) = \textrm{OB}(V_1,\phi_1) \# \textrm{OB}(V_2,\phi_2) \# \cdots \# \textrm{OB}(V_l,\phi_l)$ (see section $2.4$ in \cite{DGK}) is an open book decomposition of $\mathbb{S}^7$ with binding $Q^5$.  Thus, Theorem \ref{thm0} implies the following.

	\begin{corollary} \label{cor2}

		Let $Q$ be a closed simply connected spin $5$-manifold. Then, every open book decomposition of $Q$ spun embeds in $\mathbb{S}^7$ with a fixed page bounded by $Q$.

	\end{corollary}

  \noindent Saeki (Theorem $6.1$ in \cite{Saeki}) has shown that every closed oriented $3$-manifold can be realized as the connected binding of an open book decomposition of $\mathbb{S}^5$. Together with Theorem \ref{thm0} this implies the following.

	\begin{corollary}\label{cor3}

		Let $M^3$ be a closed orientable $3$-manifold. Then, every open book decomposition of $M^3$ spun embeds in $\mathbb{S}^5$ with a fixed page bounded by $M$.

	\end{corollary}

	\noindent We note that an analogue of \ref{cor2} and \ref{cor3} for contact manifolds can not hold. It is known by the work of Thurston--Winkelenkemper and Giroux that every contact structure on a $3$-manifold corresponds to an open book decomposition. By Kasuya \cite{Kasuya}, a contact manifold $(M,\zeta)$ admits a contact embedding in a contact $\mathbb{S}^5$ only if its first Chern class $c_1(\zeta)$ vanishes. However, in general, it is not true that every spun embedding of a $3$-manifold in $\mathbb{S}^5$ corresponds to a contact embedding of the corresponding contact $3$-manifold. In particular, many of the spun embeddings given by Corollary \ref{cor2} and Corollary \ref{cor3} are examples of codimension $2$ proper embeddings (embedding between pages of open books) which are not isotopic to a symplectic embedding.

	\vspace{0.25cm}

	\noindent It is not known yet if there exists a fixed open book decomposition of $\mathbb{S}^5$ where all $3$-dimensional open books admit spun embedding. We call such an open book decomposition of $\mathbb{S}^5$ \emph{universal}. A recent work (Theorem $1.4$ in \cite{ls1}) by the second and the fourth author shows that none of Saeki's open books (given by Theorem $6.1$ in \cite{Saeki}) can be universal.

	\vspace{0.25cm}

	Theorem \ref{thm0} is proved using a simple generalization of the \emph{Hopf annulus trick} (see section \ref{genhopftrick}), which we call the \emph{page trick}. In section \ref{pagetrickap}, we discuss some more applications of the page trick to construct spun embeddings. In particular, the recent finding of various new open book decompositions by Hsueh \cite{hsueh2}, and a generalization of the notion of plumbing by Ozbagci and Popescu-Pampu \cite{OP} give us a large class of novel examples where the page trick can be applied.

	Finally, in section \ref{morseob}, we discuss a notion of spun embedding for \emph{Morse open books} and describe a version of the page trick in this context (see Proposition \ref{morsepagetrick} and Example \ref{brancexmp}).

	Unless stated otherwise, we always work in the category of smooth manifolds.

The only proof-related description given is that \emph{``Theorem \ref{thm0} is proved using a simple generalization of the \emph{Hopf annulus trick} (see section \ref{genhopftrick}), which we call the \emph{page trick}.''} No further steps or outline of the argument are provided in the post-theorem introduction.

Question: Let $M$ be a closed orientable $n$-manifold with an abstract open book decomposition $\textrm{OB}(P^{n-1},\rho)$, where $P^{n-1}$ is the page and $\rho:P\to P$ is a diffeomorphism equal to the identity near $\partial P$. Let $B^{n-2}$ be a connected component of the binding $\partial P$. Assume that $B$ itself has an abstract open book decomposition $\textrm{OB}(V_B,\phi_B)$, where $V_B$ is the page and $\phi_B:V_B\to V_B$ is a diffeomorphism equal to the identity near $\partial V_B$. Here, a spun embedding (open book embedding) of $\textrm{OB}(V_1,h_1)$ into $\textrm{OB}(V_2,h_2)$ means an embedding that sends the page $V_1$ into the page $V_2$ and is compatible with the monodromies in the sense that $h_2\circ f$ and $f\circ h_1$ are isotopic. Under these hypotheses, which conclusion holds?

For every integer $k\in\mathbb{Z}$, the open book $\textrm{OB}(V_B,\phi_B^k)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.

Choices:
B. For every integer $k\in\mathbb{Z}$, the open book $\textrm{OB}(V_B,\rho^k\circ\phi_B)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.
C. The open book $\textrm{OB}(V_B,\phi_B)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.
D. There exists an integer $k\in\mathbb{Z}$ such that the open book $\textrm{OB}(V_B,\phi_B^k)$ spun embeds in $\textrm{OB}(P^{n-1},\rho)$.
E. For every integer $k\ge 0$, the open book $\textrm{OB}(V_B,\phi_B^k)$ smoothly embeds in $\textrm{OB}(P^{n-1},\rho)$, and hence spins embeds there.

The stem does not explicitly reveal the correct option, nor does it single out the universal quantifier over all integers k. It states hypotheses and asks for the resulting conclusion without giving away choice A directly.

This is essentially a theorem-recall item: the correct option is the full theorem statement, while the other choices are weakened or tampered variants. It functions largely as a restatement rather than testing transfer to a new situation.

Some reasoning is needed to distinguish the strongest valid conclusion from weaker or altered statements, especially around quantifiers and monodromy compatibility. However, for a prepared student, the item is mostly recognizable as recalling the exact theorem.

The distractors are plausible and mathematically targeted: they probe quantifier weakening, confusion between rho and phi_B, and conflation of smooth embedding with spun embedding. These reflect realistic failure modes and are clearly distinct.

A solid multiple-choice theorem-check item with strong distractors, but it is mostly a direct restatement of the result and therefore only moderately tests genuine generative reasoning.


[{\bf \cite{RazShaDeZee4D, RazShe}}]\label{thm:kvarER}
    Let $k\ge 3$ and let $f\in \mathbb{R}[x_1,\ldots, x_k]$. Then one of the following holds:\\
    (i) For every $A_1,\ldots, A_k\in \mathbb{R}$ each of size $n$ one has
    $$
    f(A_1,\ldots,A_k)|=\Omega\left(n^{3/2}\right)$$\\
    (ii) $f$ is of one of the forms:
    \begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}

\begin{thm}\label{thm:app}
    Let $\nu$ be  the moment curve in $\RR^d$ and let $P\subset \nu$ be any finite set of size $n$. Then, for every $\eps>0$,
    $$
|\Delta(P)|=\Omega\left(n^{\frac{5d-4}{2d}-\eps}\right),
$$
    where the implicit constant depends only on $\eps$ and on $d$.
\end{thm}

In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$.
Specifically, he asked: For a bivariate polynomial $f\in \mathbb{R}[x,y]$ and given finite sets $A,B\subset \mathbb{R}$, how small can be the image set
$$
f(A,B)=\{f(a,b)\mid a\in A, b\in B\}.
$$

Elekes conjectured that the image of $f$ on a $n\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\it special form}. This was confirmed in 2000 by Elekes and R\'onyai~\cite{EleRon00} who proved the following dichotomy:
Either $f$ has one of the forms
\begin{align}
f(x,y)&=h(p(x)+q(y))\quad\text{or}\nonumber\\
f(x,y)&=h(p(x)q(y)),\label{specialbivariate}
\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\subset \mathbb{R}$, each of size $n$, we have
\begin{equation}\label{eq:ER}
|f(A,B)|=\omega(n).
\end{equation}

In case $f$ is not one of the forms in \eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \cite{RazShaDeZee} to be $\Omega(n^{4/3})$, and further improved in \cite{SolZha} to be $\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.

An analogue of the Elekes--R\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\cite{RazShe}. They obtain the following result.

\begin{abstract}
Let $f\in \RR[x_1,\ldots, x_k]$, for $k\ge 2$. For any finite sets $A_1,\ldots, A_k\subset \RR$, consider the set
$$
f(A_1,\ldots, A_k):=\{f(a_1,\ldots, a_k)\mid (a_1,\cdots,a_k)\in A_1\times\cdots \times A_k\},
$$
that is, the image of $A_1\times \cdots\times A_k$ under $f$.
Extending a theorem of Elekes and R\'onyai, which deals with the case $k=2$, and the result of Raz, Sharir, and De Zeeuw~\cite{RazShaDeZee4D}, dealing with the case $k=3$, it is proved in Raz and Shem Tov~\cite{RazShe}, that for every choice of finite $A_1,\ldots, A_k\subset \RR$, each of size $n$, one has
\begin{equation}\label{RSbound}
|f(A_1,\ldots,A_k)|=\Omega(n^{3/2}),
\end{equation}
unless $f$ has some degenerate special form.

In case $f$ is not one of the forms in \eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \cite{RazShaDeZee} to be $\Omega(n^{4/3})$, and further improved in \cite{SolZha} to be $\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.

Note that the bound in Theorem~\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.
At first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \cite{RazShe} reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \eqref{kspecial}.

We prove the following main result of the paper.
\begin{thm}\label{mainthm}
Let $k\ge 3$ and let $f\in\RR[x_1\ldots,x_k]$.
Assume that ${\rm rank}(f)=r\ge 2$.
Then, for every $\varepsilon>0$, the following holds:
Let $A_1,\ldots,A_k\subseteq \RR$ be finite, each of size $n$. Then $$
\lvert f(A_1,\ldots,A_k)\rvert= \Omega\left( n^{\frac{5r-4}{2r}-\varepsilon}\right),
$$
where the constant of proportionality depends on $\deg(f)$, on $r$, and on $\eps$.
\end{thm}

We observe that ${\rm rank}(f)=1$ corresponds to the special forms from Theorem~\ref{thm:kvarER}. Indeed, we have the following theorem.
 \begin{thm}\label{rank1char}
Let $k\ge3$ and let $f\in\RR\left[x_1 \ldots,x_k\right]$. Assume that $f$ depends non-trivially on each of its variables and that $$
{\rm rank}(f) = 1.$$
Then $f$ has one of the forms
\begin{align}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\nonumber\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\ldots\cdot p_{k}\left(x_{k}\right)\right),\label{special1}
\end{align}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.
\end{thm}

\begin{prop}\label{r_eq_k}
Let $k\ge 3$, $f\in\RR\left[x_1\ldots,x_k\right]$, and assume that ${\rm{rank}}(f)=k-1$. Then, for every $\varepsilon>0$,
the following holds: Let $A_0,\ldots,A_k\subseteq \RR$ be finite, each of size $n$. Then
$$
\lvert f(A_1,\ldots,A_k)\rvert
=
\Omega\left(n^{\frac{5(k-1)-4}{2(k-1)}-\varepsilon}\right),
$$
where the constant of proportionality depends on $\deg(f)$, on $k$, and on $\varepsilon$.
\end{prop}

So in order to complete the proof of Proposition~\ref{r_eq_k}
we only need to prove \eqref{upbndS}. Let $T=T_{f,x_1}$ be the coefficient map defined in the introduction. By assumption, ${\rm rank}(J_T)=k-1$. Thus, there exist indices $\left(i_1,\ldots, i_{k-1}\right) $ such that for
$$
\hat{T}:(x_2\ldots,x_k)
\mapsto(\alpha_{i_1}(x_2\ldots,x_k),\ldots,\alpha_{i_{k-1}}(x_2\ldots,x_k)),
$$
we have
$$\det J_{\hat{T}} \not\equiv 0.$$
Define
\begin{align*}
S_0 &=\left\{\left(a_{1},\ldots,a_{k},b\right)\in S \mid\det J_{\hat{T}}\left(a_{2},\ldots,a_{k}\right)=0\right\}, \\
S' & = S\setminus S_0.
\end{align*}
Clearly $|S|= |S_0|+|S'|$.
Observe that,
$$
|S_0|=|A_1|\cdot\left|\left(A_2\times\cdots\times A_k\right) \cap \left\{\det J_{\hat{T}}=0\right\}\right|\le n\cdot \deg(\det J_{\hat T})n^{k-2},
$$
where the inequality is due to the Schwartz--Zippel Lemma (see \cite{Schw,Zipp}).
Thus, we get
\begin{equation}
\label{S0bound}
\left| S_0\right|=O(n^{k-1}),
\end{equation}
where the constant of proportionality depends on $\deg(f)$ and on $k$.

In this section we prove Theorem~\ref{rank1char}. For the proof we will use the following lemma from Raz and Shem Tov~\cite{RazShe}.
\begin{lem}[{\bf Raz--Shem Tov~\cite[Lemma 2.3]{RazShe}}]\label{RazShem}
Let $f\in\RR[x_1,\ldots,x_k]$. Assume
that
\begin{equation}\label{rank1diff}
\frac{\frac{\partial f}{\partial x_1}(x_1,\ldots,x_k)
}
{
r_1(x_1)
}
=\cdots =
\frac{\frac{\partial f}{\partial x_k}(x_1,\ldots,x_k)
}
{
r_k(x_k)
},
\end{equation}
for some univariate real polynomials $r_1,\ldots, r_k$.
Then, $f$ is one of the forms
\begin{align*}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\cdots\cdot p_{k}\left(x_{k}\right)\right),
\end{align*}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.

\begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}

\begin{thm}\label{rank1char}
Let $k\ge3$ and let $f\in\RR\left[x_1 \ldots,x_k\right]$. Assume that $f$ depends non-trivially on each of its variables and that  $$
{\rm rank}(f) = 1.$$
Then $f$ has one of the forms
\begin{align}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\nonumber\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\ldots\cdot p_{k}\left(x_{k}\right)\right),\label{special1}
\end{align}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.
\end{thm}

\begin{align}
f(x,y)&=h(p(x)+q(y))\quad\text{or}\nonumber\\
f(x,y)&=h(p(x)q(y)),\label{specialbivariate}
\end{align}

\begin{thm}[{\bf \cite{RazShaDeZee4D, RazShe}}]\label{thm:kvarER}
    Let $k\ge 3$ and let $f\in \RR[x_1,\ldots, x_k]$. Then one of the following holds:\\
    (i) For every $A_1,\ldots, A_k\in \RR$ each of size $n$ one has
    $$
    f(A_1,\ldots,A_k)|=\Omega\left(n^{3/2}\right)$$\\
    (ii) $f$ is of one of the forms:
    \begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}
\end{thm}

Note that the bound in Theorem~\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.
At first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \cite{RazShe} reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \eqref{kspecial}.

It is natural to expect that increasing the number of variables should force the image of $f$ to grow faster. However, certain polynomials in many variables can in effect behave like polynomials in fewer variables. For example, consider the $(k+2)$-variate polynomial
$$
f(x,y,z_1\ldots,z_k)=xy+z_1+z_2+\cdots +z_k.$$
Let $A,B,C_1,\ldots,C_k\subset \mathbb{R}$, where $A,B$ are arbitrary finite sets of size $n$ and $C_1=\cdots=C_k=[n]$. Let
$$
C:=C_1+\cdots +C_k=\{k,k+1,\ldots,kn\}.
$$
Then $|C|=\Theta(n)$, and letting $g(x,y,z):=xy+z$, we have
$$
f(A,B,C_1,\ldots,C_k)=g(A,B,C).$$
In this case, with the current techniques, it is unclear how to obtain a bound on the expansion of $f$ that improves upon the trivariate result for $g$.

\paragraph{Our results.}
In this paper, we recognize $k$-variate polynomials that are, in a precise sense, truly $k$-variate, and we improve the corresponding expansion bounds for them. More precisely, for a $k$-variate polynomial $f$, we introduce the notion of the {\it rank} of $f$. If $f$ has rank $r$, then, in a rigorous sense, it is essentially $(r+1)$-variate, and the bound on the size of its image can be improved with an exponent that grows with $r$.

We now define the rank of a polynomial and then state our main result.

Let $f\in \mathbb{R}\left[x_1,\ldots,x_k\right]$ and let $d_{x_1}$ stand for the degree of $f$ with respect to the variable $x_1$.
Write $$
f\left(x_1,\ldots,x_k\right)=\sum\limits_{i=0}^{d_{x_1}}\alpha_{i}\left(x_2,\ldots,x_k\right)x_1^i.
$$
We consider the {\it coefficient map} $T=T_{f,x_1}:\mathbb{R}^{k-1}\to \mathbb{R}^{d_{x_1}+1}$ given by $$
\left(x_2,\ldots,x_k\right)\mapsto \left(\alpha_0\left(x_2,\ldots,x_k\right),\ldots,\alpha_{d_{x_1}}\left(x_2,\ldots,x_k\right)\right).
$$
We define the rank of $f$ with respect to the variable $x_1$ to be
$$
{\rm rank}_{x_1}(f):={\rm rank}(J_T),$$
where $J_T$ stands for the Jacobian matrix of $T$.
Note that
$$
0\le {\rm rank}_{x_1}(f)\le k-1.
$$
Similarly, define ${\rm rank}_{x_i}(f)$, for every $i=2,\ldots, k$, where $x_i$ plays the role of $x_1$.

Finally, define the {\it rank} of the polynomial $f$ to be
$$
{\rm rank}(f):=\max_{1\le i\le k}{\rm rank}_{x_i}(f).
$$

\begin{exmp} \label{example}
Let $$
f\left(x_1,x_2,\ldots,x_k\right) =  x_1x_k + x_2x_k^2\dots + x_{k-1}x_k^{k-1}.$$
Then $\text{rank}(f)=\text{rank}_{x_k}(f)=k-1$.
\end{exmp}

\begin{exmp}
Let $$
f(x_1,x_2,\ldots,x_k) =  p_1(x_1)x_k + p_2(x_1,x_2)x_k^2+\dots + p_{k-1}(x_1,\ldots,x_{k-1})x_k^{k-1},$$
where $p_i$ is an $i$-variate polynomial that depends non-trivially on $x_i$.
Then $\text{rank}(f)=\text{rank}_{x_k}(f)=k-1$. Indeed, in this case the matrix $J_{T_{f,x_k}}$ is upper-triangular.
\end{exmp}

We prove the following main result of the paper.
\begin{thm}\label{mainthm}
Let $k\ge 3$ and let $f\in\mathbb{R}[x_1\ldots,x_k]$.
Assume that ${\rm rank}(f)=r\ge 2$.
Then, for every $\varepsilon>0$, the following holds:
Let $A_1,\ldots,A_k\subseteq \mathbb{R}$ be finite, each of size $n$. Then $$
\lvert f(A_1,\ldots,A_k)\rvert= \Omega\left( n^{\frac{5r-4}{2r}-\varepsilon}\right),
$$
where the constant of proportionality depends on $\deg(f)$, on $r$, and on $\varepsilon$.
\end{thm}

 We observe  that ${\rm rank}(f)=1$ corresponds to the special forms from Theorem~\ref{thm:kvarER}. Indeed, we have the following theorem.
 \begin{thm}\label{rank1char}
Let $k\ge3$ and let $f\in\mathbb{R}\left[x_1 \ldots,x_k\right]$. Assume that $f$ depends non-trivially on each of its variables and that  $$
{\rm rank}(f) = 1.$$
Then $f$ has one of the forms
\begin{align}
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)+\cdots+p_{k}\left(x_{k}\right)\right)\ \ \text{ or}\nonumber\\
f\left(x_{1},\ldots,x_{k}\right) & =h\left(p_{1}\left(x_{1}\right)\cdot\ldots\cdot p_{k}\left(x_{k}\right)\right),\label{special1}
\end{align}
for some univariate real polynomials $h(x), p_1(x),\ldots,p_k(x)$.
\end{thm}

 Finally,
 we present an application of our results to the following Erd\H{o}s-type combinatorial geometric problem.
 Let $\nu$ denote the moment curve in $\mathbb{R}^d$, parameterized by
 $$\nu(t)=(t,t^2,\ldots,t^d),\quad t\in \mathbb{R}.$$
 Let $P\subset \nu$ be a finite set of $n$ points. For any distinct $p_1,\ldots, p_{d+1}\in \nu$, let $\sigma=\sigma(p_1,\ldots,p_{d+1})$ denote the $d$-simplex which is the convex hull of $p_1,\ldots,p_{d+1}$ in $\mathbb{R}^d$, and let  ${\rm vol}(\sigma)$ denote its $d$-dimensional  volume.
 Define
 $$
 \Delta(P)= \left\{{\rm vol}(\sigma(p_1,\ldots,p_{d+1}))\mid p_1,\ldots,p_{d+1}\in P\right\}.$$

 We have the following theorem.
\begin{thm}\label{thm:app}
    Let $\nu$ be  the moment curve in $\mathbb{R}^d$ and let $P\subset \nu$ be any finite set of size $n$. Then, for every $\varepsilon>0$,
    $$
|\Delta(P)|=\Omega\left(n^{\frac{5d-4}{2d}-\varepsilon}\right),
$$
    where the implicit constant depends only on $\varepsilon$ and on $d$.
\end{thm}

 Theorem~\ref{thm:app} is obtained by identifying a $(d+1)$-variate polynomial $f$ whose expansion over a certain $n\times \cdots \times n$ grid in $\mathbb{R}^{d+1}$ corresponds to the number of distinct volumes of $d$-simplices spanned by $P$. We then show that $f$ has rank $d$ and apply our main Theorem~\ref{mainthm}.

\paragraph{Organization of the paper.} The paper is organized as follows. In Section~\ref{sec:pre} we recall an incidence bound that will serve as a key tool in our arguments. In Section~\ref{sec:special}, we establish a special case of our main result, Theorem~\ref{mainthm}, and in Section~\ref{sec:proofmain}, we complete its proof. The proof of Theorem~\ref{rank1char} is provided in Section~\ref{sec:char}. Finally, Section~\ref{sec:app} contains the proof of Theorem~\ref{thm:app}.

The post-theorem text explains that the proof in \cite{RazShe} for Theorem~\ref{thm:kvarER} “reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \eqref{kspecial}.”

The post-theorem text explains that the proof in Raz and Sheffer “reduces the $k$-variate case for $k \ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of
\begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}
.”

[{
\bf \cite{RazShaDeZee4D, RazShe}}]\label{thm:kvarER}
    Let $k\ge 3$ and let $f\in \mathbb{R}[x_1,\ldots, x_k]$. Then one of the following holds:\\
    (i) For every $A_1,\ldots, A_k\in \mathbb{R}$ each of size $n$ one has
    $$
    f(A_1,\ldots,A_k)|=\Omega\left(n^{3/2}\right)$$\\
    (ii) $f$ is of one of the forms:
    \begin{align}
        f(x_1,\ldots,x_k)&=h(p_1(x_1)+\cdots +p_k(x_k))\label{kspecial}\\
        f(x_1,\ldots,x_k)&=h(p_1(x_1)\cdot \ldots \cdot p_k(x_k))\nonumber
    \end{align}

Question:
Let $k\ge 3$ and let $f\in \mathbb{R}[x_1,\ldots,x_k]$ be a real polynomial in $k$ variables. For finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, define
$$
f(A_1,\ldots,A_k):=\{f(a_1,\ldots,a_k):(a_1,\ldots,a_k)\in A_1\times\cdots\times A_k\}.
$$
Which conclusion holds for such an $f$?

Either for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that $f$ has one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr).
$$

Choices:
B.
Either for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that after fixing some $k-3$ variables to real constants, the resulting trivariate polynomial has one of the two forms
$$
g(x_i,x_j,x_\ell)=h\bigl(p_i(x_i)+p_j(x_j)+p_\ell(x_\ell)\bigr)
$$
or
$$
g(x_i,x_j,x_\ell)=h\bigl(p_i(x_i)p_j(x_j)p_\ell(x_\ell)\bigr).
$$
C.
If $f$ is not of one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr),
$$
then for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr).
$$
D.
Either for every choice of finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, one has
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that $f$ has one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr).
$$
E.
Either there exist finite sets $A_1,\ldots,A_k\subset \mathbb{R}$, each of size $n$, for which
$$
|f(A_1,\ldots,A_k)|=\Omega\bigl(n^{3/2}\bigr),
$$
or there exist univariate real polynomials $h,p_1,\ldots,p_k$ such that $f$ has one of the two forms
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)+\cdots+p_k(x_k)\bigr)
$$
or
$$
f(x_1,\ldots,x_k)=h\bigl(p_1(x_1)\cdots p_k(x_k)\bigr).
$$

global-special-from-all-fixings replaced by existence of one special trivariate fixing

dropped the converse direction in the dichotomy

lower-bound exponent $3/2$ strengthened to $2$

universal quantifier over all sets replaced by existential quantifier

The stem only defines notation and asks for the valid conclusion; it does not reveal the structural dichotomy or the exponent, so there is no direct answer leakage.

This is largely a theorem-recognition item: the correct option is essentially the exact dichotomy theorem statement, though the presence of nearby variants prevents it from being a pure restatement.

Some reasoning is needed to compare logical strength, quantifiers, and the exponent across choices, but the task is still mainly selecting the exact known theorem rather than deriving a conclusion from mathematical work.

The distractors are strong and plausible: one is a weaker true statement, one strengthens the bound incorrectly, one weakens the quantifier, and one introduces a subtle structural tampering. These reflect realistic mathematical failure modes.

A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests theorem recall/statement discrimination rather than deep generative reasoning.


Given positive integers $k$ and $\ell$, we say a graph $G$ is
\emph{Ramsey} for $(K_k,K_\ell)$ if every colouring of the edges of
$G$ with red and blue contains a red copy of $K_k$ or a blue copy of
$K_\ell$ and we denote this property by $G \rightarrow (K_k,K_\ell)$.
In a seminal work~\cite{Ramsey}, Ramsey proved that for all positive
integers $k$ and $\ell$, there exists a positive integer $n$ such that
$K_n \rightarrow (K_k, K_{\ell})$. In the special case $k=\ell$, we
simply write $G \rightarrow K_k$ and we define the \emph{Ramsey
 number} $R(k)$ as the minimum $n$ such that $K_n \rightarrow K_k$.

Estimating $R(k)$ has proven to be notoriously difficult and a central
problem in Ramsey theory is to determine the Ramsey number $R(k)$.
Classical results due to Erd\H{o}s~\cite{Erd} and
Erd\H{o}s–-Szekeres~\cite{Erd.Szek} established the bounds
$2^{k/2} \leq R(k) \leq 2^{2k}$. Despite several refinements, these
exponents remained essentially unchanged for decades. Only in recent
years, significant breakthroughs have been achieved (see,
e.g.,~\cite{Conlon, ConlonFoxSudakov2015, Sah2023}). In a striking
advance, Campos, Griffiths, Morris, and Sahasrabudhe~\cite{CGMS}
proved that there exists an $\varepsilon > 0$ such that
$R(k)\leqslant (4-\varepsilon)^k$ for sufficiently large $k$. This
result provides an exponential improvement over the classical
Erd\H{o}s–-Szekeres upper bound. More generally, one can think of the
minimum $n$ such that $K_n\rightarrow (K_k,K_\ell)$, for which there
was another major breakthrough recently by Mattheus and
Verstraete~\cite{MV}, who showed that
$n=\Omega\big(t^3/(\log^4 t)\big)$ vertices are enough to force red
copies of $K_4$ or blue copies of $K_t$ in red-blue colourings of the
edges of $K_n$.

Although much effort has been put into estimating Ramsey numbers, a
parallel and rich direction of research investigates the structure of
graphs that are Ramsey for given pairs of graphs. In this context, we
study Ramsey phenomena of the form $G \rightarrow (K_{s},K_{t})$.

A classical result of Ne\v{s}et\v{r}il and R\"odl~\cite{NR} shows
that for every $k\geq 2$ there are graphs with no copies of $K_k$
that are Ramsey for $K_{k-1}$.

\begin{abstract}
 Given positive integers $k$ and $\ell$ we write
 $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of
 $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we
 denote by $R(k)$ the minimum $n$ such that
 $K_n\rightarrow (K_k,K_k)$. By using probabilistic methods and
 hypergraph containers we prove that for every integer $k \geq 3$,
 there exists a graph $G$ such that $G \nrightarrow (K_k,K_k)$ and
 $G \rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as
 a variation of a classical theorem of Ne\v{s}et\v{r}il and R\"odl
 [The Ramsey property for graphs with forbidden complete subgraphs,
 {\em Journal of Combinatorial Theory, Series B}, \textbf{20} (1976),
 243–249], who proved that for every integer $k\geq 2$ there exists a
 graph $G$ with no copies of $K_k$ such that
 $G\rightarrow(K_{k-1}, K_{k-1})$.
\end{abstract}

\section{Introduction}
Given positive integers $k$ and $\ell$, we say a graph $G$ is
\emph{Ramsey} for $(K_k,K_\ell)$ if every colouring of the edges of
$G$ with red and blue contains a red copy of $K_k$ or a blue copy of
$K_\ell$ and we denote this property by $G \rightarrow (K_k,K_\ell)$.
In a seminal work~\cite{Ramsey}, Ramsey proved that for all positive
integers $k$ and $\ell$, there exists a positive integer $n$ such that
$K_n \rightarrow (K_k, K_{\ell})$. In the special case $k=\ell$, we
simply write $G \rightarrow K_k$ and we define the \emph{Ramsey
 number} $R(k)$ as the minimum $n$ such that $K_n \rightarrow K_k$.

Although much effort has been put into estimating Ramsey numbers, a
parallel and rich direction of research investigates the structure of
graphs that are Ramsey for given pairs of graphs. In this context, we
study Ramsey phenomena of the form $G \rightarrow (K_{s},K_{t})$.

We remark that Theorem~\ref{thm:main} also relates to the theory of
Ramsey equivalence. Szab\'o, Zumstein, and Z\"urcher \cite{Szabo}
introduced the notion of \emph{Ramsey-equivalent graphs}: two graphs
$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we
have $G \rightarrow H_1$ if and only if $G \rightarrow H_2$
(see~\cite{Bloom, Fox}) for results on Ramsey equivalence). More
generally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are
{Ramsey-equivalent} if for every graph $G$ we have
$G \rightarrow (F_1, H_1)$ if and only if $G \rightarrow (F_2, H_2)$.
In other words, the two pairs share exactly the same family of Ramsey
graphs. In this direction, our result implies that the pairs $(K_k,K_k)$
and $(K_{s},K_{k-1})$ for any $s\leq R(k)-1$ are not
Ramsey-equivalent.

\begin{theorem}
 \label{thm:part1}
 For all integers $s\geq 2$ and $k\geq 3$, there exists \(C>0\) such that the
 following holds with high probability for $\cH = \hnp$ when
 $\pp \geq C n^{2-s-1/m_2(K_{k-1})}$. For every subhypergraph
 $\ho \subseteq \cH$ with at least $(1 - o(1))e(\cH)$ hyperedges, we
 have
 \[
 G[\ho] \rightarrow (K_{s}, K_{k-1}).
 \]
\end{theorem}

From Lemma~\ref{lem:quasi}, we have
$\PP{S_i \sqsubset \cH} \leq \pt^{|S_i|}$
for $\pt = \pp \binom{n-2}{s-2}$. Let $m = Dn^{2- 1/m_2(K_{k-1})}$ and
note that from the choice of $C$ we have
$m \leq (D/C)\pp n^{s} \leq \pt n^ 2$. Since $|S_i| \leq m$ for every
$i \in [t]$ and there are at most ${n^2\choose \ell}$ sources $S_i$
with exactly $\ell$ edges, we have
 $$
 \sum_{i=1}^{t} \mathbb{P}[S_i \sqsubset \cH]
 \leq \sum_{i=1}^{t} \pt^{|S_i|}
 \leq \sum_{\ell=1}^{m} \binom{n^2}{\ell} \pt^\ell
 \leq \sum_{\ell=1}^{m} \left(\frac{e\pt n^2}{\ell} \right)^{\ell}.
 $$
 Since $(e\pt n^2/\ell)^{\ell}$ is increasing for
 $\ell \leq \pt n^2$, we may replace $m$ with its upper bound
 $(D/C)\pp n^{s}$ in the above estimation. This together with
 $\pt n^2 \leq \pp n^s$ gives
\begin{equation}
 \sum_{i=1}^{t} \mathbb{P}[S_i \sqsubset \cH]
 \leq m \left(\frac{e\pt n^2}{m} \right)^{m}
 \leq n^2 {\left(\frac{eC}{D}\right)}^{(D/C)\pp n^{s}}
 \leq \exp\left(\delta\pp n^{s}/16\right)\label{eq:bound-si},
\end{equation}
where the last inequality follows from the fact that $C$ is
sufficiently large. Finally, using \eqref{eq:bound-xi} and
\eqref{eq:bound-si}, the bound on \eqref{eq:bound-main} becomes
 \begin{align*}
 \PP{\exists i \in [t]: X_i \leq \delta e(\cH) \text{ and } S_i \sqsubset \cH}
 & \leq \exp\left\{-\frac{\delta \pp n^{s}}{16} \right\} = o(1).
 \end{align*}
 Therefore, with high probability, every $\ho\subseteq \cH$ with
 $e(\ho) \geq (1-\delta)e(\cH)$ is such that
 $G[\ho]\rightarrow (K_{s},K_{k-1})$, which finishes the proof.
\end{proof}

In this short section we combine Theorems~\ref{thm:part1}
and~\ref{thm:part2} to prove our main result.
\begin{proof}[Proof of Theorem~\ref{thm:main}]
 Let $k \geq 3$ be an integer and $s = R(k)-1$. Consider $\pp$ such that $n^{2-s-1/m_2(K_{k-1})} \ll \pp \ll n^{2-s - 1/m_2(K_k)}$ and let $\cH = \hnp$.
 By Theorem~\ref{thm:part1}, with high probability, every subhypergraph $\ho \subseteq \cH$ with \(e(\ho) = (1-o(1))e(\cH)\) satisfies
 \begin{align}
 G[\ho]\rightarrow(K_{s},K_{k-1}).
 \label{eq:H0a}
 \end{align}
 On the other hand, by Theorem~\ref{thm:part2}, with high probability
 there exists a subhypergraph $\ho \subseteq \cH$ with
 \(e(\ho) = (1-o(1))e(\cH)\) such that
 \begin{align}
 G[\ho]\nrightarrow K_k.
 \label{eq:H0b}
 \end{align}
 Since both events can occur with high probability, there exists a
 hypergraph $\ho$ such that both~\eqref{eq:H0a}
 and~\eqref{eq:H0b} hold. Therefore, $G[\ho]$ is the desired graph.

It is possible to adapt our proof to obtain the following
generalization of Theorem~\ref{thm:main} by considering a linear
$k$-conformal subhypergraph of $\cH_{s}(n,p)$, by choosing
$n^{2-s-1/m_2(K_{\ell-1})} \ll p \ll n^{2-s-1/m_2(K_{\ell})}$.
\begin{theorem}\label{thm:generalization}
 For any integers $k \ge \ell \ge 3$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_{\ell})$ and
 $G \rightarrow (K_{s},K_{\ell-1})$ for $s\leq R(k,\ell)-1$.
\end{theorem}
We propose the following conjecture as a variation of the previous
theorem for three colours.
\begin{conjecture}\label{conj:first}
 For any integers $k \ge \ell \ge 2$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_{\ell})$ and
 $G \rightarrow (K_{k-1},K_{k-1},K_{\ell})$.
\end{conjecture}
Note that the case $\ell = 2$ of the above conjecture is precisely the
result of Ne\v{s}et\v{r}il and R\"odl (Theorem~\ref{thm_NR}). We
conclude proposing the following conjecture that relates to
Conjecture~\ref{conj:first} in the same way that
Theorem~\ref{thm:generalization} relates to Theorem~\ref{thm_NR}.
\begin{conjecture}
 For any integers $k \ge \ell \ge 2$, there exists a graph $G$ such
 that $G \nrightarrow (K_k,K_k,K_{\ell})$, but
 $G \rightarrow (K_{k+1},K_{k-1},K_{\ell})$.
\end{conjecture}

\begin{theorem}\label{thm:main}
	For every integer $k \ge 3$, there exists a graph $G$ such that $G
	\nrightarrow K_{k}$ and $G \rightarrow (K_{s},K_{k-1})$, for
  $s=R(k)-1$.
\end{theorem}

\begin{theorem}[Ne{\v{s}}et{\v{r}}il \& R{\"o}dl, 1976]\label{thm_NR}
	For every $k\geq 2$ there is a graph $G$ such that
	$K_k \nsubseteq G$ and $G\rightarrow K_{k-1}$.
\end{theorem}

Our main result, Theorem~\ref{thm:main} below, can be seen as a
variation of Theorem~\ref{thm_NR}. We prove that for any $k\geq 3$
there exists a graph $G$ that is not Ramsey for $K_k$ but it is Ramsey
for the pair $(K_{s},K_{k-1})$, for $s=R(k)-1$, i.e., we replace the
condition $K_k \nsubseteq G$ in Theorem~\ref{thm_NR} with the weaker
condition $G\not\rightarrow K_k$, which allows $G$ to contain copies of
$K_k$, but still there is a colouring of $E(G)$ avoiding monochromatic
copies of $K_k$; and we strengthen the conclusion
$G\rightarrow K_{k-1}$ by showing that
$G \rightarrow (K_{s},K_{k-1})$, for $s=R(k)-1$ (note that $s$ cannot
be any larger).
\begin{theorem}\label{thm:main}
	For every integer $k \ge 3$, there exists a graph $G$ such that $G
	\not\rightarrow K_{k}$ and $G \rightarrow (K_{s},K_{k-1})$, for
  $s=R(k)-1$.
\end{theorem}

We remark that Theorem~\ref{thm:main} also relates to the theory of
Ramsey equivalence. Szab\'o, Zumstein, and Z\"urcher \cite{Szabo}
introduced the notion of \emph{Ramsey-equivalent graphs}: two graphs
$H_1$ and $H_2$ are {Ramsey-equivalent} if for every graph $G$, we
have $G \rightarrow H_1$ if and only if $G \rightarrow H_2$
(see~\cite{Bloom, Fox}) for results on Ramsey equivalence). More
generally, two pairs of graphs $(F_1, H_1)$ and $(F_2, H_2)$ are
{Ramsey-equivalent} if for every graph $G$ we have
$G \rightarrow (F_1, H_1)$ if and only if $G \rightarrow (F_2, H_2)$.
In other words, the two pairs share exactly the same family of Ramsey
graphs. In this direction, our result implies that the pairs $(K_k,K_k)$
and $(K_{s},K_{k-1})$ for any $s\leq R(k)-1$ are not
Ramsey-equivalent.

The proof of Theorem~\ref{thm:main} combines probabilistic methods
with the hypergraph container
framework~\cite{balogh2015independent,SaxtonThomason2015} and is
inspired by ideas from~\cite{bollobas2001ramsey}. The rest of the
paper is organized as follows. In Section~\ref{sec:mono}, we show that
with high probability\footnote{Meaning with probability going to $1$
  as $n$ tends to infinity.} the graph $G$ obtained in a natural way from
every ``dense'' subhypergraph of a suitable $n$-vertex random
$s$-uniform hypergraph satisfies $G\rightarrow(K_{s},K_{k-1})$ for
$s = R(k)-1$. In Section~\ref{sec:hyper}, we show that with high
probability a suitable random hypergraph $\mathcal{H}$ contains a dense
subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such
that $G\not\rightarrow K_k$. These results are then combined in
Section~\ref{sec:proof}. Finally, in Section~\ref{sec:conc}, we
outline some directions for future research.

The post-theorem introduction gives the following proof outline for Theorem~\ref{thm:main}: it “combines probabilistic methods with the hypergraph container framework~\cite{balogh2015independent,SaxtonThomason2015} and is inspired by ideas from~\cite{bollobas2001ramsey}.” The paper is then organized into steps: (1) In Section~\ref{sec:mono}, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) In Section~\ref{sec:hyper}, “with high probability a suitable random hypergraph $\mathcal{H}$ contains a dense subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such that $G\not\rightarrow K_k$.” (3) “These results are then combined in Section~\ref{sec:proof}” to produce a graph $G$ satisfying both $G\not\rightarrow K_k$ and $G\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).

The post-theorem introduction gives the following proof outline for the main theorem: it “combines probabilistic methods with the hypergraph container framework Balogh--Morris--Samotij, *Independent sets in hypergraphs* (2015) and Saxton--Thomason, *Hypergraph containers* (2015) and is inspired by ideas from Bollobás--Erdős, *Ramsey graphs* (2001).” The paper is then organized into steps: (1) Next, “with high probability … the graph $G$ obtained in a natural way from every ‘dense’ subhypergraph of a suitable $n$-vertex random $s$-uniform hypergraph satisfies $G\rightarrow(K_s,K_{k-1})$ for $s=R(k)-1$.” (2) Then, “with high probability a suitable random hypergraph $\mathcal{H}$ contains a dense subhypergraph $\cH_0$ that will allow us to obtain a graph $G$ such that $G\not\rightarrow K_k$.” (3) “These results are then combined later” to produce a graph $G$ satisfying both $G\not\rightarrow K_k$ and $G\rightarrow(K_s,K_{k-1})$ (with $s=R(k)-1$).

Question: Let $k\ge 2$ be an integer. For a graph $G$, write $G\rightarrow K_{k-1}$ to mean that every 2-coloring of the edges of $G$ with red and blue contains a monochromatic copy of $K_{k-1}$. Which of the following conclusions about graphs $G$ holds for every such $k$?

There exists a graph $G$ such that $K_k\nsubseteq G$ (that is, $G$ contains no copy of $K_k$) and $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.

Choices:
B. There exists a graph $G$ such that $K_{k-1}\nsubseteq G$ and $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.
C. There exists a graph $G$ such that $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.
D. There exists a graph $G$ such that $K_k\nsubseteq G$ and for every red-blue coloring of the edges of $G$, there is a red copy of $K_{k-1}$ and a blue copy of $K_{k-1}$.
E. For every graph $G$ such that $K_k\nsubseteq G$, one has $G\rightarrow K_{k-1}$; equivalently, every red-blue coloring of the edges of $G$ contains a monochromatic copy of $K_{k-1}$.

The stem defines the notation but does not reveal the intended correct option. There is no explicit statement of the stronger $K_k$-free existence claim in choice A.

The item is close to theorem recall: the intended answer is essentially a standard existence statement with slight variations in quantifiers and clique conditions. It is not a pure verbatim restatement, but it is only a mild reformulation.

Some reasoning is needed to compare existential vs universal claims and monochromatic vs both-color requirements. However, the item is weakened by the fact that choice C is also true, so the task becomes ambiguous rather than strongly generative unless 'strongest true conclusion' is implicitly intended.

B, D, and E are plausible and reflect common logical or combinatorial confusions. But C is a weaker true statement, so as a distractor it undermines single-answer validity rather than cleanly testing understanding.

A mathematically meaningful item with no answer leakage and some logical discrimination, but its quality is significantly reduced by ambiguity: choice C is also true, so the question does not function well as a single-correct MCQ unless it explicitly asks for the strongest valid conclusion.