The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,
irreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are "irreducible" for the bi-action.\\
This result is well-known for groups of finite Morley rank \cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\
An interesting family of theories is finite-dimensional theories, in the sense of \cite{wagner2020dimensional}.
\begin{defn}
    A theory $T$ is said to be \emph{finite-dimensional} if there exists a function $\operatorname{dim}$ from the class of all interpretable subset in any model $\mathcal{M}$ of $T$ into $\omega\cup\{-\infty\}$ such that for any $\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,
\begin{itemize}
    \item If $a,a'$ have same type over $\emptyset$, $\operatorname{dim}(\phi(x,a))=\operatorname{dim}(\phi(x,a'))$
    \item $\operatorname{dim}(\emptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
    \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
    \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
    \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
    \item $\Gamma,\Delta$ sharply commute;
    \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
    \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
    There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
    Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
    \begin{itemize}
        \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
        \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
        \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
        \item $A_0=\Kat(\Gamma,\Delta)$.
    \end{itemize}
    Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

mptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
 \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
 \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
 \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
 \item $\Gamma,\Delta$ sharply commute;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
 There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
 Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
 \begin{itemize}
 \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
 \item $A_0=\Kat(\Gamma,\Delta)$.
 \end{itemize}
 Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

\begin{lemma}\label{boundedind}
    Let $G$ be a definable group of finite dimension and $\{H_i\}_{i\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\omega$ such that there is no $J=\{j_1,...,j_n\}\subseteq I$ of cardinality $n$ such that $|\bigcap_{i=1}^kH_{j_i}:\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\leq n-1$.
\end{lemma}

\def\notind#1#2{#1\setbox0=\hbox{$#1x$}\kern\wd0
\hbox to 0pt{\mathchardef\nn=12854\hss$#1\nn$\kern1.4\wd0\hss}
\hbox to 0pt{\hss$#1\mid$\hss}\lower.9\ht0 \hbox to 0pt{\hss$#1\smile$\hss}\kern\wd0}

\end{theorem}
\begin{proof}
 We prove the base case. Given $\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\Delta$, that therefore is finite.\\
 Being the katakernel finite, there exists only boundedly many elements for $\Delta$ or $\Delta$ is unbounded (and then the proof follows using the previous idea).\\
 Then we can take a representative with maximal image and we have to prove that
\end{proof}
\begin{theorem}

\begin{proof}
Let $n=\dim(G)$. We will work in a sufficiently saturated structure $\mathfrak{M}$ in which $G$ is definable and assume that $\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\in \omega$, there exists $g_1,...,g_N$ such that $|\bigcap_{j\leq k-1} \phi(\mathfrak{M},g_j)/\bigcap_{j\leq k} \phi(\mathfrak{M},g_j)|\geq k$. Therefore, the partial type given by the formulas
$$\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\in \bigcap_{j=1}^i \phi(\mathfrak{M},x_j)\wedge a^i_j{a^i_k}^{-1}\not\in \phi(\mathfrak{M},x_j)\wedge \forall_{i\leq N}\ \phi(\mathfrak{M},x_i)\leq G(\mathfrak{M})$$
for all $k<\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\{H_i\}_{i\leq N}$ such that $|\bigcap_{i<j}H_i/\bigcap_{i\leq j}H_i|$ is infinite for every $i$. This would imply that the dimension of $G$ is strictly greater than $n$, a contradiction.
\end{proof}
If we apply these results to the centralizers of elements in $G$, we obtain that a finite-dimensional definable group has the hereditary $\widetilde{\mathfrak{M}}_c$-property.
\begin{defn}
    A group $G$ is \emph{hereditarily-$\widetilde{\mathfrak{M}}_c$} if for any definable subgroups $H,N$, such that $N$ is normalised by $H$, there exist natural numbers $n_{HN}$ and $d_{HN}$ such that any sequence of centralisers
    $$C_H(a_0/N)\geq C_H(a_0,a_1/N)\geq...\geq C_H(a_0,a_1,...,a_n/N)\geq...$$
    with each centraliser of index at least $d_{HN}$ in the previous one, has length at most $n_{HN}$.
\end{defn}
The notions of almost containment and commensurability are fundamental in the study of groups definable in finite-dimensional theories.
\begin{defn}
\begin{itemize}
    \item  Let $G$ be a group and $A, B\leq G$ subgroups. $A$ \emph{almost contains} $B$, denoted $A\apprge B$, if $|B:B\cap A|$ is finite.
    \item  If $A\apprle B$ and $B\apprle A$, $A$ and $B$ are \emph{commensurable}.
    \item Given a family of subgroups $\{G_i\}_{i\in I}$ in $G$, the family $\{G_i\}_{i\in I}$ is \emph{uniformly commensurable} if there exists $n<\omega$ such that $|G_i/G_i\cap G_j|\leq n$ for every $i,j\in I$.
\end{itemize}
  \end{defn}

Assume towards a contradiction that the conclusion of Theorem~\ref{boundedind} fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. Hence the desired bounds $n,d<\omega$ exist.

Assume towards a contradiction that the conclusion of the main theorem fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. This completes the proof of the main theorem.

Question: Let $G$ be a definable group of finite dimension, and let $\{H_i\}_{i\in I}$ be a uniformly definable family of subgroups of $G$ (that is, all the subgroups $H_i$ are given by instances of a single definable formula). Which statement holds about finite chains of successive intersections of members of this family?

There exist $n,d<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.\] Equivalently, there do not exist $n$ distinct indices $j_1,\dots,j_n\in I$ such that each successive index \(\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\) is greater than $d$ for every $k\le n-1$.

Choices:
B. There exists $d<\omega$ such that for every $n<\omega$ and every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.\] Equivalently, one may choose a single bound $d$ working uniformly for chains of all finite lengths.
C. There exists $n<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|<\infty.\]
D. There exist $n,d<\omega$ such that for every sequence of distinct indices $j_1,\dots,j_n\in I$ and every $k\le n-1$, one has \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.\] In other words, all successive indices along every chain of length $n$ are uniformly bounded by the same finite constant.
E. There exists $n<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\bigcap_{i=1}^k H_{j_i}=\bigcap_{i=1}^{k+1} H_{j_i}.\] Equivalently, in every chain of $n$ successive intersections, one inclusion must already be equality.

dropped the uniform finite bound d, retaining only finiteness of one successive index

bounded index confused with equality of successive intersections


For every pair of fixed integers $r,\ell\geqslant 3$, consider the random greedy high girth $r$-clique
removal process on vertex set $[n]$. Let $M$  be the number of edges in the
generated linear $r$-graph with girth larger than $\ell$ when the process terminates.
With high probability, there exists some positive constant $\lambda=\lambda(\ell)$  such that
\begin{align*}
M\geqslant n^{1+ \frac{1}{\ell-1}-\lambda \frac{\log\log n}{\log n}}.
\end{align*}

For an integer $r\geqslant 3$,
 a hypergraph $H=(V_H,E_H)$ on vertex set $[n]$ is an
 \textit{$r$-uniform hypergraph} (\textit{$r$-graph} for short) if each edge
  is an $r$-element subset (\textit{$r$-set} for short) of $[n]$,
 and is said to be \textit{linear} if any two distinct edges share at most
 one vertex. Write ${v}_H$ and ${e}_H$ to denote the number of vertices and
 edges of $H$. An $r$-graph $H$ is called a partial Steiner $(n,r,s)$-system
 for an integer $s$ with $2\leqslant s\leqslant r-1$,
 if every $s$-set lies in at most one
 edge of it. In particular,  partial Steiner $(n,r,2)$-systems are
 linear $r$-graphs, partial Steiner $(n,3,2)$-systems are partial
 Steiner triple systems. Given a family $\mathcal{H}$ of linear $r$-graphs, define the
\textit{linear Turán number} of $\mathcal{H}$, denoted by
 $\textmd{ex}_r^L(n,\mathcal{H})$, is the maximum
number of edges among  linear $r$-graphs on vertex set $[n]$ that
 contain no $r$-graphs of $\mathcal{H}$ as subgraphs ($\mathcal{H}$-\textit{free} for short).
 Recently, the research on the Turán problem in linear hypergraphs has attracted
 considerable attention. Let
  $\textmd{Forb}_r^L(n,\mathcal{H})$ be the set
 of all  linear $\mathcal{H}$-free $r$-graphs on
vertex set $[n]$. When $\mathcal{H}$ consists of a single linear
 $r$-graph $H$, we simply write them
as $\textmd{ex}_r^L(n,{H})$  and
$\textmd{Forb}_r^L(n,{H})$ instead. The Tur\'{a}n problem and the enumeration problem
for a family $\mathcal{H}$ of $r$-graphs are closely related. Every subgraph of a $\mathcal{H}$-free graph
is also $\mathcal{H}$-free, then a trivial bound on the size of  $\textmd{Forb}_r^L(n,\mathcal{H})$
is given in~\cite{balgoh17} as follows
\begin{align*}
2^{\textmd{ex}_r^L(n,\mathcal{H})}\leqslant |\textmd{Forb}_r^L(n,\mathcal{H})|\leqslant
\sum_{i\leqslant \textmd{ex}_r^L(n,\mathcal{H})}\binom{\binom{n}{r}}{i}\leqslant 2n^{r\cdot \textmd{ex}_r^L(n,\mathcal{H})}.
\end{align*}

 There are several different notions of hypergraph cycles.
  For an integer $\ell\geqslant 3$, a Berge cycle $B_\ell^r$
  of length $\ell$ is an alternating sequence of distinct vertices and distinct
  edges of the form ${v}_1,\boldsymbol{e}_1,{v}_2,\boldsymbol{e}_2,\ldots,{v}_\ell,\boldsymbol{e}_\ell$,
  where ${v}_i,{v}_{i+1}\in \boldsymbol{e}_i$ for each $i\in[\ell-1]$ and ${v}_\ell,{v}_1\in \boldsymbol{e}_\ell$.
 This definition
  of a hypergraph cycle is the classical definition due to Berge.
    Note that forbidding a Berge cycle $B_\ell^r$ actually forbids a family of hypergraphs.
    Another notion of hypergraph cycles that has been actively investigated   is that of a linear cycle, which is unique up to isomorphism. An $r$-uniform linear cycle  $C_\ell^r$ of length $\ell$
 is an alternating sequence ${v}_1,\boldsymbol{e}_1,{v}_2,\boldsymbol{e}_2,\ldots,{v}_\ell,\boldsymbol{e}_\ell$ of distinct vertices and distinct
  edges such that $\boldsymbol{e}_i\cap \boldsymbol{e}_{i+1}=\{{v}_{i+1}\}$ for each $i\in[\ell-1]$,
   $\boldsymbol{e}_1\cap \boldsymbol{e}_\ell=\{{v}_1\}$ and $\boldsymbol{e}_i\cap \boldsymbol{e}_j=\emptyset$ if $1<|j-i|<\ell-1$.
The \textit{girth} of a linear $r$-graph is to be the smallest integer $\ell$
such that it contains a $C_\ell^r$. If no such $\ell$ exists,
the girth of the linear $r$-graph is defined to be infinite.

Determining $\textmd{ex}_3^L(n,C_3^3)$ is equivalent to the
famous $(6,3)$-problem, which aims at determining the maximum number of edges
of triple systems not carrying three edges on six vertices. The $(6,3)$-problem is studied
by Brown, Erd\H{o}s and S\'{o}s~\cite{brown73}, and it is known that for some constant $c>0$,
\begin{align*}
n^{2- {c}{\sqrt{\log n}}}<\textmd{ex}_3^L(n,C_3^3)=o(n^2),
\end{align*}
where the lower bound is given by Behrend~\cite{behrend46} and the upper bound is given by
Rusza and Szemer\'{e}di~\cite{ruz78}.
Erd\H{o}s, Frankl and R\"{o}dl~\cite{erd86}
showed that for every $r\geqslant 3$, $\textmd{ex}_r^L(n,C_3^r)=o(n^2)$ and
$\textmd{ex}_r^L(n,C_3^r)>n^{2-o(1)}$.
Using the so-called $2$-fold Sidon sets,
Lazebnik and  Verstra\"{e}te~\cite{laz03} constructed linear $3$-graphs with girth $5$
and showed that
\begin{align*}
\textmd{ex}_3^L\bigl(n,\{B_3^3, B_4^3\}\bigr)= \frac{1}{6}n^{  \frac{3}{2}}+O(n).
\end{align*}
Kostochka, Mubayi and Verstra\"{e}te asked if for integers $r\geqslant 3$ and $\ell\geqslant 4$,
\begin{align*}
\textmd{ex}_r^L\bigl(n,C_\ell^r\bigr)=\Theta\bigl(n^{1+ \frac{1}{\lfloor \ell/2\rfloor}}\bigr).
\end{align*} Later, Collier-Cartaino, Graber and Jiang~\cite{coll14}
proved that $\textmd{ex}_r^L(n,C_\ell^r)=O(n^{1+ \frac{1}{\lfloor \ell/2\rfloor}})$.
 In detail,
they obtained there exist positive constants $c(r, k)$ and $d(r, k)$
 such that
 \begin{align*}
 \textmd{ex}_r^L(n,C_{2k}^r)\leqslant c(r,k)n^{1+ \frac{1}{k}}\ \text{and }
 \textmd{ex}_r^L(n,C_{2k+1}^r)\leqslant d(r,k)n^{1+ \frac{1}{k}},
 \end{align*}
where the integer $k\geqslant 2$, the constants $c(r, k)$ and $d(r, k)$ are
exponential in $k$ for fixed $r$.
As an immediate corollary of the main results in~\cite{jiang30}, Jiang, Ma and Yepremyan
further improved the coefficient $c(r, k)$ to be linear in $k$.
Ergemlidze et al.~\cite{ergem2019} strengthened some results in $3$-graphs to be
$\textmd{ex}_3^L(n, B_4^3)= \frac{1}{6}n^{  3/2}+O(n)$, $\textmd{ex}_3^L(n, B_5^3)= \frac{1}{3\sqrt{3}}n^{  3/2}+O(n)$,
and for $k=2,3,4$ and $6$,
\begin{align*}
\textmd{ex}_3^L\bigl(n,\{C_3^3,C_5^3,\ldots,C_{2k+1}^3\}\bigr)= \Theta\bigl(n^{1+ \frac{1}{k}}\bigr).
\end{align*}
In general,  for $k\geqslant 2$, they also showed that \begin{align*}
\textmd{ex}_3^L\bigl(n,\{C_3^3,C_5^3,\ldots,C_{2k+1}^3\}\bigr)= \Omega\bigl(n^{1+ \frac{2}{3k-4+\epsilon}}\bigr),
\end{align*}
where $\epsilon=0$ if $k$ is odd and $\epsilon=1$ if $k$ is even.
As a special case of the linear hypergraph extension of K\H{o}v\'{a}ri–S\'{o}s–Tur\'{a}n's theorem in~\cite{gao2021},
Gao and Chang showed that $\textmd{ex}_3^L(n, C_4^3)= \frac{1}{6}n^{  3/2}+O(n)$,
and Gao et al. in~\cite{gao2023} further proved that $\textmd{ex}_3^L(n,C_5^3)= \frac{1}{3\sqrt{3}}n^{  3/2}+O(n)$.
For the lower bound of $\textmd{ex}_r^L(n,C_\ell^r)$, the best known lower bound  for $r=3$ is \begin{align*}
\textmd{ex}_3^L\bigl(n,C_\ell^3\bigr)=\Omega(n^{1+ \frac{1}{\ell-1}})
\end{align*}
that was shown in~\cite{coll14,ergem2019} due to Verstra\"{e}te, by taking a random subgraph of a Steiner
triple system.
Using generalized Sidon sets~\cite{ergem2019}, it is conceivable that one can obtain a similar (or
better) constructive lower bound of $\textmd{ex}_3^L(n,C_\ell^3)$
(and maybe also for all $r\geqslant 3$), which is an area worth some exploration.
The lower bound on the linear Tur\'{a}n number of linear cycles is still
far from what is conjectured, especially in higher uniformities $r\geqslant 4$.
Sometimes to explore the lower bound of the corresponding
Tur\'{a}n problem is more difficult than the upper one.

Following the same logic with the Tur\'{a}n problems of cycles,
for integers $r\geqslant 3$ and $\ell\geqslant 4$, Balogh and Li~\cite{balgoh17} conjectured that
$ |\textmd{Forb}_r^L(n,C_\ell^r)|=2^{\Theta(n^{1+1/\lfloor \ell/2\rfloor})}$.
They confirmed the conjecture in the case of $\ell=4$.
For $\ell>4$, they provided a result on the girth version, that is,
there exists a constant $c=c(r,\ell)$ such that
the number of linear $r$-graphs with girth larger than $\ell$ is at most
$2^{c\cdot n^{1+1/\lfloor \ell/2\rfloor}}$.
Define $\textmd{ex}_L(n,r,\ell)$ and $\textmd{Forb}_L(n,r,\ell)$
as  $\textmd{ex}_r^L(n,\mathcal{H})$ and $\textmd{Forb}_r^L(n,\mathcal{H})$ of
$\mathcal{H}=\{C_i^r:\, 3\leqslant i\leqslant \ell\}$, respectively.
Hence, it implies that $|\textmd{Forb}_L(n,r,\ell)|=2^{O(n^{1+1/\lfloor \ell/2\rfloor})}$, while its lower bound remains open.
The idea of the proof in~\cite{balgoh17} is to reduce the
 hypergraph enumeration problems to some graph enumeration problems, and
then the graph container method is implemented in it.
Balogh, Narayanan and Skokan~\cite{bal18} indeed made use of
the hypergraph container method and provided a balanced supersaturation theorem
for linear cycles to  resolve a conjecture on the
enumeration of $r$-uniform hypergraphs with a forbidden $C_\ell^r$.

In order to solve a conjecture of Erd\H{o}s about the existence
of sparse Steiner triple systems,
Glock, K\"{u}hn, Lo and Osthus~\cite{glock20}, and
Bohman and Warnke~\cite{boh19}, studied a constrained triple process along
similar lines, which can be
seen as a generalization of the \textit{random greedy triangle removal process}~\cite{bohman15},
to iteratively build a sparse partial Steiner triple system,
that is,
adding the sparseness constraint does not affect
the evolution of the process significantly.
To be precise, a Steiner triple system is called $k$-sparse if it does not
contain any $(i+2,i)$-configurations for integers $2\leqslant i\leqslant k$,
where a $(i+2,i)$-configuration
is a set of $i$ triples  which span at most $i+2$ vertices.
Erd\H{o}s made his conjecture
in Steiner triple systems.
Some further problems to
higher uniformities have been proposed
and considered in~\cite{fure13,glock20}.

The
 \textit{random greedy $r$-clique removal process} starts with a complete graph
 on vertex set $[n]$, denoted as $\mathbb{G}(0)$,
 and $\mathbb{G}(i+1)$ is the remaining graph from $\mathbb{G}(i)$ by
 selecting one $r$-clique uniformly at random out of all $r$-cliques in $\mathbb{G}(i)$
 and deleting all its edges from the edge set ${E}(i)$ of $\mathbb{G}(i)$.
 The process terminates once the remaining graph contains no $r$-cliques,
 and along the way produces a linear $r$-graph, where it starts with the
 empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each
 chosen $r$-clique as a new edge in the hypergraph.
 Let $M=\min\{i\,:\mathbb{G}(i)\ {\mbox{is\ } r{\mbox {-clique free}}}\}$,
 and  ${E}(M)$ be the set of edges left unsaturated
 by the produced $r$-cliques.  In particular, this problem attracted much  attention when $r=3$ and it is also called the random greedy triangle removal process.
Fewer results are known on $r\geqslant 4$.
Bennett and Bohman~\cite{bennett15} considered the random greedy
hypergraph matching process, which can be seen as a generalization
of the $r$-clique removal process.
Tian et al.~\cite{tian23} directly discussed the structure
of the random greedy $r$-clique removal process to a natural barrier,
 using a heuristic assumption to find the trajectories of an ensemble
 of random variables when the process evolves.
Inspired by the above works, for integers $r\geqslant 3$ and $\ell\geqslant 3$,
we consider the constrained {random greedy $r$-clique removal process}
so that it
does not just produce a linear $r$-graph but with girth larger than $\ell$,
 and the corresponding process is called  the \textit{random greedy high girth $r$-clique removal process}
 or \textit{high girth linear $r$-uniform hypergraph process}.
Compared with their analysis in~\cite{glock20,boh19} discussing sparse Steiner triple systems,
the challenge
arising in the analysis of random greedy high girth $r$-clique removal process is that each individual
forbidden configuration $C_i^r$ with $3\leqslant i\leqslant \ell$ should be considered.
It is more sensitive and stricter to establish
 the trajectories and error functions of an ensemble
 of random variables which we track in this constrained process with a higher uniformity,
such that the one-step
 changes, trend hypotheses and boundedness hypotheses of these tracked variables can  support the proof about the concentration of all these variables to the end.
 Finally, we show the following theorem.

Following the same logic with the Tur\'{a}n problems of cycles,
for integers $r\geqslant 3$ and $\ell\geqslant 4$, Balogh and Li~\cite{balgoh17} conjectured that
$ |\textmd{Forb}_r^L(n,C_\ell^r)|=2^{\Theta(n^{1+1/\lfloor \ell/2\rfloor})}$.
They confirmed the conjecture in the case of $\ell=4$.
For $\ell>4$, they provided a result on the girth version, that is,
there exists a constant $c=c(r,\ell)$ such that
the number of linear $r$-graphs with girth larger than $\ell$ is at most
$2^{c\cdot n^{1+1/\lfloor \ell/2\rfloor}}$.
Define $\textmd{ex}_L(n,r,\ell)$ and $\textmd{Forb}_L(n,r,\ell)$
as $\textmd{ex}_r^L(n,\mathcal{H})$ and $\textmd{Forb}_r^L(n,\mathcal{H})$ of
$\mathcal{H}=\{C_i^r:\, 3\leqslant i\leqslant \ell\}$, respectively.
Hence, it implies that $|\textmd{Forb}_L(n,r,\ell)|=2^{O(n^{1+1/\lfloor \ell/2\rfloor})}$, while its lower bound remains open.
The idea of the proof in~\cite{balgoh17} is to reduce the
 hypergraph enumeration problems to some graph enumeration problems, and
then the graph container method is implemented in it.
Balogh, Narayanan and Skokan~\cite{bal18} indeed made use of
the hypergraph container method and provided a balanced supersaturation theorem
for linear cycles to resolve a conjecture on the
enumeration of $r$-uniform hypergraphs with a forbidden $C_\ell^r$.

In order to solve a conjecture of Erd\H{o}s about the existence
of sparse Steiner triple systems,
Glock, K\"{u}hn, Lo and Osthus~\cite{glock20}, and
Bohman and Warnke~\cite{boh19}, studied a constrained triple process along
similar lines, which can be
seen as a generalization of the \textit{random greedy triangle removal process}~\cite{bohman15},
to iteratively build a sparse partial Steiner triple system,
that is,
adding the sparseness constraint does not affect
the evolution of the process significantly.
To be precise, a Steiner triple system is called $k$-sparse if it does not
contain any $(i+2,i)$-configurations for integers $2\leqslant i\leqslant k$,
where a $(i+2,i)$-configuration
is a set of $i$ triples which span at most $i+2$ vertices.
Erd\H{o}s made his conjecture
in Steiner triple systems.
Some further problems to
higher uniformities have been proposed
and considered in~\cite{fure13,glock20}.

The
 \textit{random greedy $r$-clique removal process} starts with a complete graph
 on vertex set $[n]$, denoted as $\mathbb{G}(0)$,
 and $\mathbb{G}(i+1)$ is the remaining graph from $\mathbb{G}(i)$ by
 selecting one $r$-clique uniformly at random out of all $r$-cliques in $\mathbb{G}(i)$
 and deleting all its edges from the edge set ${E}(i)$ of $\mathbb{G}(i)$.
 The process terminates once the remaining graph contains no $r$-cliques,
 and along the way produces a linear $r$-graph, where it starts with the
 empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each
 chosen $r$-clique as a new edge in the hypergraph.
 Let $M=\min\{i\,:\mathbb{G}(i)\ {\mbox{is\ } r{\mbox {-clique free}}}\}$,
 and ${E}(M)$ be the set of edges left unsaturated
 by the produced $r$-cliques. In particular, this problem attracted much attention when $r=3$ and it is also called the random greedy triangle removal process.
Fewer results are known on $r\geqslant 4$.
Bennett and Bohman~\cite{bennett15} considered the random greedy
hypergraph matching process, which can be seen as a generalization
of the $r$-clique removal process.
Tian et al.~\cite{tian23} directly discussed the structure
of the random greedy $r$-clique removal process to a natural barrier,
 using a heuristic assumption to find the trajectories of an ensemble
 of random variables when the process evolves.
Inspired by the above works, for integers $r\geqslant 3$ and $\ell\geqslant 3$,
we consider the constrained {random greedy $r$-clique removal process}
so that it
does not just produce a linear $r$-graph but with girth larger than $\ell$,
 and the corresponding process is called the \textit{random greedy high girth $r$-clique removal process}
 or \textit{high girth linear $r$-uniform hypergraph process}.
Compared with their analysis in~\cite{glock20,boh19} discussing sparse Steiner triple systems,
the challenge
arising in the analysis of random greedy high girth $r$-clique removal process is that each individual
forbidden configuration $C_i^r$ with $3\leqslant i\leqslant \ell$ should be considered.
It is more sensitive and stricter to establish
 the trajectories and error functions of an ensemble
 of random variables which we track in this constrained process with a higher uniformity,
such that the one-step
 changes, trend hypotheses and boundedness hypotheses of these tracked variables can support the proof about the concentration of all these variables to the end.
 Finally, we show the following theorem.

In order to solve a conjecture of Erd\H{o}s about the existence
of sparse Steiner triple systems,
Glock, K\"{u}hn, Lo and Osthus~\cite{glock20}, and
Bohman and Warnke~\cite{boh19}, studied a constrained triple process along
similar lines, which can be
seen as a generalization of the \textit{random greedy triangle removal process}~\cite{bohman15},
to iteratively build a sparse partial Steiner triple system,
that is,
adding the sparseness constraint does not affect
the evolution of the process significantly.
To be precise, a Steiner triple system is called $k$-sparse if it does not
contain any $(i+2,i)$-configurations for integers $2\leqslant i\leqslant k$,
where a $(i+2,i)$-configuration
is a set of $i$ triples which span at most $i+2$ vertices.
Erd\H{o}s made his conjecture
in Steiner triple systems.
Some further problems to
higher uniformities have been proposed
and considered in~\cite{fure13,glock20}.

The remainder of the paper is structured as follows.
Notation and auxiliary
results used throughout the paper are presented in Section 2.
We define the random greedy high girth $r$-clique
removal process in Section 3, introducing some key random variables of the process that we wish to track,
estimating the corresponding expected trajectory and
 choosing error function for each tracked variable.
 We formally prove the concentration of all these variables in Section 4,
 where the required one-step changes, trend hypotheses and boundedness hypotheses for
these tracked variables are analyzed in higher uniformities.
 Two important claims to bound the overcounting and boundedness parameters
 are discussed in detail in the final section.

The \textit{random greedy high girth $r$-clique removal process} starts with a complete graph
 on vertex set $[n]$, denoted by $\mathbb{G}(0)$,
 and $\mathbb{G}(i+1)$ is the remaining graph from $\mathbb{G}(i)$ by
 selecting one $r$-clique uniformly at random out of all \textit{available} $r$-cliques in $\mathbb{G}(i)$
 and deleting all its edges from the edge set ${E}(i)$ of $\mathbb{G}(i)$,
 where an $r$-clique is called \textit{available} in $\mathbb{G}(i)$ means that adding
 the vertex set of this $r$-clique as an edge
 does not produce an $r$-graph in $\mathcal{L}$ with the edges generated by
 the vertices of previously chosen available $r$-cliques.
 Let ${M}=\min\{i:\mathbb{G}(i) {\text{ is} \textit{ available } r{\text{-}\textit{clique} \text{ free}}}\}$.
 This process is equivalently viewed as creating
 a random linear $r$-graph with girth greater than $\ell$, that is, the
 \textit{random greedy high girth linear $r$-uniform hypergraph process}.
 Beginning with the empty $r$-graph $\mathbb{H}(0)$ on the same vertex set $[n]$ we
sequentially set $\mathbb{H}({i+1})=\mathbb{H}(i)+\boldsymbol{e}_{i+1}$, where the added $r$-set $\boldsymbol{e}_{i+1}$
is the vertex set of the chosen available $r$-clique in $\mathbb{G}(i)$, and
$\boldsymbol{e}_{i+1}$ is also called to be an available $r$-set or available edge for $\mathbb{H}(i)$.
 The process terminates at a maximal linear $r$-graph $\mathbb{H}({M})$ with girth larger than $\ell$
 and no available $r$-sets or available edges. At time $i$, graphs are defined with
 respect to edge sets on the vertex set $[n]$, where
 both ${\mathbb{G}}(i)$ and $\mathbb{H}(i)$ depend on the outcomes of the process up
 to this point, and ${\mathbb{G}}(i)$ is the graph given by the pairs of vertices
 that do not appear in any edge of $\mathbb{H}(i)$.
It will be convenient to study the relation
 between $\mathbb{G}(i)$ and $\mathbb{H}(i)$ via
\begin{align}
{E}(i)=\binom{[n]}{2}\Bigl\backslash \bigcup_{\boldsymbol{f}\in \mathbb{H}(i)}
\biggl\{\bigl\{x,y\bigr\}\in \binom{\boldsymbol{f}}{2}\biggr\}.
\end{align}

\begin{theorem}
Given any $\mu>0$ and fixed integers $r,\ell\geqslant 3$, there exist $\lambda>0$ and $\alpha\in (0,1)$
such that, for any ${\boldsymbol{f}}_m\in {K}_m(i)$ with $2\leqslant m\leqslant r-1$,
${\boldsymbol{f}}\in {Q}(i)$, $L\in \mathcal{L}$ and $0\leqslant k\leqslant e_L-2$, with probability at least $1-n^{-\mu}$,
\begin{align}
|{Q}(i)|&={q}(t)\pm\epsilon_{q}(t),\\
|{Y}_{{\boldsymbol{f}}_m}(i)|&={y}_m(t)\pm \epsilon_{{y}_m}(t),\\
|{W}_{{\boldsymbol{f}},L,k}(i)|&={w}_{L,k}(t)\pm \epsilon_{w_{L,k}}(t),
\end{align}
holding when $i$ satisfying $0\leqslant i\leqslant {M}$
with ${M}$ defined as
\begin{align}{M}=n^{1+ \frac{1}{\ell-1}-\lambda \frac{\log\log n}{\log n}}.
\end{align}
Here, for $t\in [0,t_M]$,
\begin{align}{q}(t)&= \frac{n^r}{r!}p_i^{\binom{r}{2}}\xi_i,\\
{y}_m(t)&= \frac{n^{r-m}}{(r-m)!}p_i^{ \binom{r}{2}- \binom{m}{2}}\xi_i,\\
{w}_{L,k}(t)&= \frac{r!e_L}{|\texttt{Aut}(L)|}\binom{e_L-1}{k}(r!t)^{k}(p_i^{\binom{r}{2}}\xi_i)^{e_L-1-k}n^{(r-1)(e_L-k)+k-r},
\end{align}
and \begin{align}
\epsilon_{q}(t)&=\sigma_i n^{\alpha+r-1},\\\epsilon_{y_m}(t)&=\sigma_i n^{\alpha+r-m-1},\\\epsilon_{w_{L,k}}(t)&=\beta_k\sigma_i^{k+2} n^{\alpha+(r-1)(e_L-k)+k-r-1}t_{\textmd{M}}^k,
\end{align}in which \begin{align}
\sigma_i=\sigma(t)=\log(n^\alpha+n^2t),
\end{align} $\beta_k$ is a constant depending on $k$ and
\begin{align}
t_{{M}}= \frac{{M}}{n^2}=n^{-1+ \frac{1}{\ell-1}-\lambda \frac{\log\log n}{\log n}}.
\end{align}
\end{theorem}

Theorem~3.4 is proved in Section 4 and we
always tacitly assume $0\leqslant i\leqslant {M}$ with ${M}$ defined in~(3.18).
 It implies that the process produces
a linear $r$-graph of size at least ${M}$ with its girth greater than $\ell$.
Theorem~3.4 verifies Theorem~1.2
with room to spare in the power of the logarithmic factor.
We make no attempt to optimize the constants $\lambda$, $\alpha$ and
the coefficients $\beta_k$ for any $L\in \mathcal{L}$
 and $0\leqslant k\leqslant e_L-2$.
There are many choices of
them that can be balanced to satisfy certain inequalities.
For example, for given positive integers $r$, $\ell$ and $k$, we
choose these constants to satisfy the equations \begin{align}
\lambda> \frac{\ell}{\ell-1},\quad\alpha_0<\alpha<1 \quad\text{and}\quad\beta_k= \Bigl(\frac{3\ell r!}{p_{{M}}^{ \binom{r}{2}}\xi_{{M}}}\Bigr)^{k},
\end{align}
where $\alpha_0= \frac{\ell-2}{\ell-1}$, $p_{{M}}$ and $\xi_{{M}}$ are constants
when $i=M$ defined in~(3.5) and~(3.9). We will see these choices are sufficient for our proof of Theorem 3.4 in next section.
 We do not replace them with their actual values, which is for the interest of
understanding the role of these constants played in the calculations.

\noindent  In the following proof of Theorem~1.1, we will show the term $O({\log\log n}/{\log n})$
 is necessary in the exponent of
 the lower bound of
 $M$. Furthermore, it is impossible to apply this
 approach to obtain a better lower bound than $\Omega(n^{1+ \frac{1}{\ell-1}})$.
 As a corollary of Theorem 1.1, it is natural to  obtain lower bounds
of $\textmd{ex}_L(n,r,\ell)$ and $|\textmd{Forb}_L(n,r,\ell)|$
when  $\ell\geqslant 3$.
\begin{corollary} For every pair of fixed integers $r,\ell\geqslant 3$,
there exists $\lambda=\lambda(\ell)>0$ such that as $n\rightarrow\infty$,  \begin{align*}
\textmd{ex}_L(n,r,\ell)> n^{1+ \frac{1}{\ell-1}- \frac{\lambda\log\log n}{\log n}}\quad \text{and}\quad
|\textmd{Forb}_L(n,r,\ell)|> 2^{n^{1+ \frac{1}{\ell-1}- \frac{\lambda\log\log n}{\log n}}}.
\end{align*}
\end{corollary}
\noindent

The remainder of the paper is structured as follows.
Notation and  auxiliary
results used throughout the paper are presented in Section 2.
We define the random greedy high girth $r$-clique
removal process in Section 3, introducing some key random variables of the process that we wish to track,
estimating the corresponding expected trajectory and
 choosing error function for each tracked variable.
 We formally prove the concentration of all these variables in Section 4,
 where the required one-step changes, trend hypotheses and boundedness hypotheses for
these tracked variables are analyzed in higher uniformities.
    Two important claims to bound the overcounting and boundedness parameters
    are discussed in detail in  the final section.

We define the random greedy high girth $r$-clique removal process (Section 3) and introduce “some key random variables of the process that we wish to track,” then estimate “the corresponding expected trajectory” and “choos[e] error function for each tracked variable.” Next, we “formally prove the concentration of all these variables” (Section 4) by analyzing “the required one-step changes, trend hypotheses and boundedness hypotheses for these tracked variables … in higher uniformities.” Finally, “two important claims to bound the overcounting and boundedness parameters are discussed in detail in the final section.” The proof also notes that the exponent’s term $O({\log\log n}/{\log n})$ is “necessary,” and that “it is impossible to apply this approach to obtain a better lower bound than $\Omega(n^{1+ \frac{1}{\ell-1}})$.”

Existential–Universal
Inequality or Bound

Question: For fixed integers $r,\ell\ge 3$, consider the random greedy high girth $r$-clique removal process on vertex set $[n]$: starting from the complete graph on $[n]$, one repeatedly chooses an $r$-clique uniformly at random among the currently available choices subject to the condition that the $r$-sets chosen so far form a linear $r$-uniform hypergraph of girth greater than $\ell$, adds the chosen $r$-set as a hyperedge, and deletes all graph edges of that clique; the process stops when no further such $r$-clique can be chosen. Let $M$ denote the number of hyperedges in the final generated linear $r$-graph. Which statement is guaranteed with high probability (that is, with probability tending to $1$ as $n\to\infty$)?

There exists a positive constant $\lambda=\lambda(\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}-\lambda\frac{\log\log n}{\log n}}$.

Choices:
B. There exists a positive constant $\lambda=\lambda(\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}+\lambda\frac{\log\log n}{\log n}}$.
C. There exists a positive constant $\lambda=\lambda(\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}-\lambda}$.
D. There exists a positive constant $\lambda=\lambda(r,\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}-\lambda\frac{\log\log n}{\log n}}$.
E. With high probability, for every positive constant $\lambda=\lambda(\ell)$ one has $M\ge n^{1+\frac{1}{\ell-1}-\lambda\frac{\log\log n}{\log n}}$.

necessary negative logarithmic correction

replaced the sharper $\frac{\log\log n}{\log n}$ loss by a fixed constant loss in the exponent

dependence of the constant on parameters; theorem states $\lambda=\lambda(\ell)$, not on $r$

existential choice of $\lambda$ replaced by universal quantification over all positive constants


\begin{theorem}\label{thm1}
There exists a constant $C>1$ such that
\begin{equation}\label{equnthm1}
C^{-1}\lambda^{d_S/2}\leq{N}(u)\leq C\lambda^{d_S/2}
\end{equation}
holds for any global Dirichlet or Neumann eigenfunction $u$  on $\mathcal {SG}$ with  eigenvalue $\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.
\end{theorem}

\label{sec1}
The study of Laplacians on fractals and their spectral properties constitutes a cornerstone of analysis on fractals, a field largely pioneered by Kigami \cite{K2,K4}. Unlike their Euclidean counterparts, Laplacians on fractals exhibit a wealth of novel and unexpected phenomena, leading to a rich and distinct spectral theory \cite{FS, Ka1,Ka2,K3,KL, Sh, Sh2, T}. Among these phenomena, the intricate oscillatory behavior of eigenfunctions — such as the distribution and growth of their local extrema — remains a topic of profound interest.

In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \cite{Ho} show that the $L^\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.

The landscape on fractals is quite different. The existence of pre-localized eigenfunctions — eigenfunctions that vanish identically on the boundary along with their normal derivatives — is a hallmark of fractals \cite{K3,K4}. Such eigenfunctions can be highly localized and give rise to  a cascade of new eigenfunctions through a localization process. This structure fundamentally alters the asymptotic distribution of eigenvalues and the qualitative behavior of eigenfunctions. Consequently, classical tools and intuition from elliptic PDEs often fail, necessitating new frameworks for understanding the fine properties of eigenfunctions on fractals.

\begin{figure}[h]
\begin{center}
\includegraphics[width=5.5cm]{SG2.pdf}
\caption{The Sierpinski gasket $\mathcal{SG}$.}
\label{figure1}
\end{center}
\end{figure}

This paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\mathcal{SG}$, see Figure \ref{figure1}) serves as a typical  example. We introduce and investigate the extremum counting function
$N(x)$ for the Laplacian $-\Delta_\mu$ where $\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\lambda$, the count
$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:

$$N(x)=\sup\{N(u): u \text{ is a }\lambda \text{-eigenfunction with } 0\leq \lambda\leq x\}.$$

Our primary goal is to establish the asymptotic growth rate of $N(x)$ as $x\to \infty$, linking it directly to the spectral exponent $d_S$ of the fractal, which characterizes a sharp phase transition in the behavior of $N(x)$.

For the upper bound, we prove in Proposition \ref{prothm1} that under a natural condition (A) — namely, that eigenfunctions with sufficiently small eigenvalues possess at most one extreme set — the growth of $N(x)$ is at most of the order $x^{d_S/2}$:
$$\limsup_{x\to\infty} \frac{N(x)}{x^{d_S/2}}<\infty.$$
Condition (A) can be interpreted as a form of ``low-energy simplicity''. It is not universal for all fractals, as evidenced by the modified Koch curve \cite{M,Sh2}, where even low-energy eigenfunctions can have infinitely many extrema. Verifying its validity is therefore a key step in the analysis for a given fractal.

Conversely, for the lower bound, Proposition \ref{prothm2} shows that the existence of  a  pre-localized eigenfunction forces \( N(x) \) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:
\[
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\]
The proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.

Our main result  is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\mathcal{SG}$.

In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \cite{Ho} show that the $L^\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.

\begin{figure}[h]
\begin{center}
\includegraphics[width=5.5cm]{SG2.pdf}
\caption{The Sierpinski gasket $\mathcal{SG}$.}
\label{figure1}
\end{center}
\end{figure}

This paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\mathcal{SG}$, see Figure \ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function
$N(x)$ for the Laplacian $-\Delta_\mu$ where $\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\lambda$, the count
$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:

$$N(x)=\sup\{N(u): u \text{ is a }\lambda \text{-eigenfunction with } 0\leq \lambda\leq x\}.$$

Conversely, for the lower bound, Proposition \ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \( N(x) \) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:
\[
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\]
The proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.

Our main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\mathcal{SG}$.

\begin{definition}\label{def1}

(a).
Let $u\in\mathscr{D}_\mu$, $\emptyset\neq A\subset K$ and $c\in\mathbb{R}$, we say that $u$ has a \textit{local maximum ({\rm resp.} minimum)} at $A$ with value $c$ if

(a-1). $A$ is a connected component of $u^{-1}(c)$;

(a-2). $A\cap V_0=\emptyset$;

(a-3). there exists $\delta>0$ such that $u(p)\leq c$ (resp. $u(p)\geq c$) for any $ p\in A_\delta$, where
$A_\delta$ is the $\delta$-neighborhood of $A$.

Call such a set $A$ a \textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\it extreme point} of $u$.

(b). For $u\in\mathscr{D}_\mu$, denote
\[{N}(u)=\#\{A\subset K: A \text{ is an extreme set of } u\}.\]
For $x>0$, define
\[N(x)=\sup\{{N}(u): u \text { is a $\lambda$-eigenfunction with}\ 0\leq \lambda\leq x\},\]
and call it \textit{the extremum counting function} of $-\Delta_\mu$.
\end{definition}

\begin{proposition}\label{prothm2}
If there exists a pre-localized eigenfunction, then
\begin{equation*}
\liminf_{x\to\infty}\frac{\log N(x)}{\log x}\geq\kappa
\end{equation*}
for some $0<\kappa\leq d_S/2$.
In particular, for the lattice case (see the exact meaning in Section \ref{sec2}),
\begin{equation*}
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\end{equation*}
\end{proposition}

\noindent{\bf Remark.}
For a $\lambda$-eigenfunction with $\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).
Indeed, suppose that $u$ has a local maximum at $A$ with value $c\leq 0$. Since $u$ is a $\lambda$-eigenfunction, there exists $\delta>0$ such that $u\leq c\leq 0$ and $\Delta_\mu u=-\lambda u\geq 0$ on $A_\delta$. Moreover, $u$ is non-constant on $A_\delta$. Let $h$ be the harmonic function satisfying $h|_{\partial A_\delta}=u|_{\partial A_\delta}$, and denote by $g(\cdot,\cdot)$ the Green function for $-\Delta_\mu$ on $A_\delta$. It is known that $g(p,q)>0$ for all $p,q\in A_\delta$. Then, for $p\in A$, we have
\begin{equation*}
u(p)=\int_{A_\delta}g(p,q)\lambda u(q)\mathrm{d}\mu(q)+h(p)<h(p)\leq \max u|_{\partial A_{\delta}},
\end{equation*}
which yields a contradiction.

\noindent{\bf Remark.} By Lemma \ref{pro1}-(a), $\mathbf{A}_\lambda$ and $\mathbf{B}_\lambda^1$ are cones, i.e. if $\mathbf{a}\in \mathbf{A}_\lambda\ (\text{or }\mathbf{B}_\lambda^1)$, then $k\mathbf{a}\in \mathbf{A}_\lambda \ (\text{or }\mathbf{B}_\lambda^1)$ for any $k\neq 0$. Let $\pi:\mathbb{R}^L\setminus\{\mathbf{0}\}\to\mathbb{R}\mathrm{P}^{L-1}$ be the canonical projection that sends each point to the line through the origin it generates.
For $\mathbf{a},\mathbf{b}\in\mathbb{R}^L$, define $\mathbf{a}\sim\mathbf{b}$ if there exists $k\in\mathbb{R}\setminus\{0\}$ such that $\mathbf{a}=k\mathbf{b}$, then $\pi$ can be regarded as a bijection from $(\mathbb{R}^L\setminus\{\mathbf{0}\})/\sim$ to $\mathbb{R}\mathrm{P}^{L-1}$.
Write \[\mathcal{A}_\lambda=\pi(\mathbf{A}_\lambda),\quad \mathcal{B}_\lambda^1=\pi(\mathbf{B}_\lambda^1),\] and define
\begin{equation}\label{defabT}
\mathcal{T}_\lambda^i:\mathbb{R}\mathrm{P}^{L-1}\to\mathbb{R}\mathrm{P}^{L-1} \quad \text{by}\quad \mathcal{T}_\lambda^i=\pi\circ T_\lambda^i\circ\pi^{-1}.
\end{equation}
Then the condition (\ref{eq1}) is equivalent to
\begin{equation}\tag{\ref{eq1}*}\label{eq2}
\begin{aligned}
&\mathcal{A}_\lambda=\big(\bigcup_{i\in S}(\mathcal{T}_\lambda^i)^{-1}(\mathcal{A}_{\gamma^2_i\lambda})\big)\cup\mathcal{B}^1_\lambda,\quad\text{with}\\
(\mathcal{T}_\lambda^i&)^{-1}(\mathcal{A}_{\gamma^2_i\lambda}),\ i\in S\text{ and }\mathcal{B}^1_\lambda\text{ pairwise disjoint.}
\end{aligned}
\end{equation}
Consequently, condition (A) is equivalent to \eqref{eq2}.

First we prove \eqref{exC}, i.e. $\mathcal{C}_{\lambda}=\mathcal{D}_{\lambda_0}$. Noticing that for any $\mathbf{b}\in\mathbb{R}^3\setminus\{{\bf 0}\}$, by \eqref{eq4.4}, we have
\begin{equation}\label{pixxi}
\pi(\{\mathbf{x}\in\mathbb{R}^3\setminus\{{\bf 0}\}:\mathbf{b}^\mathrm{t}\mathbf{x}=0\})=\Big\{\boldsymbol{\xi}\in\mathbb{R}\mathrm{P}^2:\ \mathbf{b}^\mathrm{t}(\mathbf{1},Q)^{-\mathrm{t}}\begin{pmatrix}1\\ \boldsymbol{\xi}\end{pmatrix}=0\Big\},
\end{equation}
and a direct calculation gives
\begin{equation*}
(4-\lambda_0,-2,-2)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,0,2(6-\lambda_0)\big).
\end{equation*}
So by Lemma \ref{pro43}, we have
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(1)}=0\}\Big)\\
=&\Big\{\boldsymbol{\xi}\in\mathbb{R}\mathrm{P}^2:\ \big(-\lambda_0,0,2(6-\lambda_0)\big)\begin{pmatrix}1\\ \boldsymbol{\xi}\end{pmatrix}=0\Big\}\\
=&\Big\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ \xi^{(2)}-\frac{\lambda_0}{2(6-\lambda_0)}=0\Big\}\cup\{[(1,0)^\mathrm{t}]_\infty\}.
\end{aligned}
\end{equation*}
Similarly, $(-2,4-\lambda_0,-2)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,-\sqrt{3}(6-\lambda_0),-(6-\lambda_0)\big)$ gives
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(2)}=0\}\Big)\\
=&\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ \sqrt{3}\xi^{(1)}+\xi^{(2)}+\frac{\lambda_0}{6-\lambda_0}=0\}\cup\{[(1,-\sqrt{3})^\mathrm{t}]_\infty\},
\end{aligned}
\end{equation*}
and $(-2,-2,4-\lambda_0)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(-\lambda_0,\sqrt{3}(6-\lambda_0),-(6-\lambda_0)\big)$ gives
\begin{equation*}
\begin{aligned}
&\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}\Big(\{(x^{(1)},x^{(2)},x^{(3)})^\mathrm{t}\in\mathbb{R}^3\setminus\{{\bf 0}\}:x^{(3)}=0\}\Big)\\
=&\{(\xi^{(1)},\xi^{(2)})^\mathrm{t}\in\mathbb{R}^2:\ -\sqrt{3}\xi^{(1)}+\xi^{(2)}+\frac{\lambda_0}{6-\lambda_0}=0\}\cup\{[(1,\sqrt{3})^\mathrm{t}]_\infty\}.
\end{aligned}
\end{equation*}
Recall that $\mathbf{C}=\{(a_1,a_2,a_3)\in\mathbb{R}^3:\text{either }a_i>0\text{ for all }i, \text{ or }a_i<0\text{ for all }i\}$ and $\mathcal{C}_\lambda=\pi\circ\tau_\lambda^D\circ(\tau_\lambda^N)^{-1}(\mathbf{C})$. By the symmetry of $\mathcal{SG}$, we have $\boldsymbol{\theta}\in\mathcal{C}_\lambda$. So the above calculation gives
$\mathcal{C}_{\lambda}=\mathcal{D}_{\lambda_0}$.

(b). For $5^{-m}\lambda<\lambda_1^D$ with $m\geq 1$, it is easy to see that
$\lambda_{m}=\psi^{-1}(5^{-m}\lambda)\in(0,6)$ by Remark \ref{re33}. Considering the function $v=u\circ F_2F_3^{m-1}$, by (\ref{gamma-delta01pre}), (\ref{gamma-delta02pre}) and $p_{23}=F_2F_3^{m-1}p_3\in A$, we see
\begin{equation}\label{gamma-delta0102r}
\begin{aligned}
&2v(p_1)+2v(p_2)-(4-\lambda_m)v(p_3)=0,\\
&\big(v(p_{3})-v(p_2)\big)\big(v(p_{3})-v(p_1)\big)\geq 0.
\end{aligned}
\end{equation}
Using (\ref{pixxi}) we find that
\begin{equation*}
\begin{aligned}
&\tau(v)\in\pi\circ\tau_\lambda^D\Big(\big\{v\in E(5^{-m}\lambda)\setminus\{0\}: v\text{ satisfies (\ref{gamma-delta0102r})}\big\}\Big)\\
=&\ \big\{(\xi^{(1)},\xi^{(2)})\in\mathbb{R}^2:\xi^{(2)}=\sqrt{3}\xi^{(1)}-\frac{\lambda_m}{6-\lambda_m},\ 0\leq\xi^{(1)}\leq\frac{\sqrt{3}\lambda_m}{2(6-\lambda_m)}\big\}\\
=&\ \mathcal{I}_{\lambda_m,3}\cup\{\boldsymbol{\zeta}_{\lambda_m,23}\}\cup\{\boldsymbol{\zeta}_{\lambda_m,31}\},
\end{aligned}
\end{equation*}
since
\begin{equation*}
\begin{aligned}
&(2,2,-(4-\lambda_m))(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(\lambda_m,-\sqrt{3}(6-\lambda_m),6-\lambda_m\big),\\
&(0,-1,1)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(0,2\sqrt{3},0\big),\\
&(-1,0,1)(\mathbf{1},Q)^{-\mathrm{t}}=\frac{1}{3}\big(0,\sqrt{3},-3\big),
\end{aligned}
\end{equation*}
and $\pi\circ \tau_\lambda^D(v')=\frac{\lambda_m}{4(6-\lambda_m)}(\sqrt{3},-1)^\mathrm{t}\in\mathcal{I}_{\lambda_m,3}$ for a function $v'\in E(5^{-m}\lambda)$ satisfying (\ref{gamma-delta0102r}) with $v'(p_1)=v'(p_2)=4-\lambda_m$ and $v'(p_3)=4$.

(b). for $u$ in the (D5) or (N5) case, there exists a constant $c_5>1$ such that
\begin{equation}\label{esti5}
c_5^{-1}\lambda^{d_S/2}\leq N(u)\leq c_5\lambda^{d_S/2};
\end{equation}

\begin{lemma}\label{lm62}
Let $u\in E_D(\lambda)\setminus\{0\}$ or $E_N(\lambda)\setminus\{0\}$ with $\mathrm{supp}\,u=\mathcal{SG}$. For $u$ in the (D6) or (N6) case and $\eps=\varnothing$, we have
\begin{equation}\label{esti6}
c_6^{-1}\lambda^{d_S/2}\leq N(u)\leq c_6\lambda^{d_S/2}
\end{equation}
for some constant $c_6>1$.
\end{lemma}
\begin{proof}
For $u$ in the (D6) or (N6) case, we have $\lambda_m=3$, $\lambda_{m-1}=6$ and $\lambda=5^m\psi(3)$. Now we consider $u\circ F_{w'}$ for each $w'\in W_{m-2}$. Since $-\Delta_{m-1}u|_{V_{m-1}}=\lambda_{m-1}u|_{V_{m-1}}$, for distinct $i,j,k\in S$ we have
\begin{equation*}
\begin{aligned}
-2u(p_{ij}^{w'})=u(p_{i}^{w'})+u(p_{j}^{w'})+u(p_{ki}^{w'})+u(p_{jk}^{w'}),
\end{aligned}
\end{equation*}
and solving the above equations gives $u(p_i^{w'})=-u(p_{ki}^{w'})-u(p_{ij}^{w'})$. Without loss of generality, we assume $u(p_{23}^{w'})\leq u(p_{31}^{w'})\leq u(p_{12}^{w'})$, then

The result demonstrates the highly regular behavior of  $\mathcal{SG}$, where eigenfunction complexity — quantified by the count of local extrema — grows  precisely as the  power law  $d_S/2$. The proof relies on a detail analysis on the eigenfunction decimation, originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.

In the classical setting, the number of critical points of an eigenfunction on an $n$-dimensional manifold is generally expected to grow linearly with $\lambda^{(n-1)/2}$, a rate related to the wave propagation and the Bohr-Sommerfeld quantization. However, in highly  symmetric setting such as $n$-dimensional rectangles or balls — where eigenfunctions can oscillate at high frequencies independently along different directions — this order improves to $\lambda^{n/2}$.

On fractals, as shown  in Theorem \ref{thm1} and Propositions \ref{prothm1} and \ref{prothm2}, the growth is governed by the spectral exponent $d_S$, which takes the role played by the geometric Hausdorff dimension $n$ in the Euclidean setting. The observed order $d_S/2$ for $\mathcal{SG}$ suggests that its high symmetry is a key factor in realizing this maximal growth rate. We therefore conjecture that for a broader class of p.c.f. self-similar sets, the growth order of local extrema is at most $d_S/2$, with this upper bound being attained only in highly symmetric cases  — such as nested fractals, or even beyond the p.c.f. setting, as exemplified by the  Sierpinski carpet — and strictly smaller in the presence of lower symmetry.

\subsection{Notation and Propositions \ref{prothm1} and \ref{prothm2}}Before ending this section, let us introduce the exact definition of   $N(u)$ and $N(x)$  on a p.c.f. self-similar set $K$, which is always assumed to be  connected.

Let $V_0$ denote the boundary of $K$, and $\mathscr{D}_\mu$ denote the domain of $\Delta_\mu$. For a function $u\in \mathscr{D}_\mu$ and $p\in V_0$, denote by $(du)_p$ the normal derivative of $u$ at $p$.
For $\lambda\geq 0$, call a non-trivial  function $u\in\mathscr{D}_\mu$ satisfying $-\Delta_\mu u=\lambda u$ on $K\setminus V_0$ a {\it $\lambda$-eigenfunction} of $-\Delta_\mu$.  Say an eigenfunction $u$ a \textit{Dirichlet (Neumann) eigenfunction} if  $u|_{V_0}=0$ ($du|_{V_0}=0$). In particular, say $u$ a \textit{pre-localized eigenfunction}  if  both $u|_{V_0}=0$ and $du|_{V_0}=0$ hold; and a \textit{global eigenfunction} if $\mathrm{supp}\,u=K$.

\begin{definition}\label{def1}

(a).
Let $u\in\mathscr{D}_\mu$, $\emptyset\neq A\subset K$ and $c\in\mathbb{R}$, we say that $u$ has a \textit{local maximum ({\rm resp.} minimum)} at $A$ with value $c$ if

(a-1). $A$ is a connected component of $u^{-1}(c)$;

(a-2). $A\cap V_0=\emptyset$;

(a-3). there exists $\delta>0$ such that $u(p)\leq c$ (resp. $u(p)\geq c$) for any $ p\in A_\delta$, where
$A_\delta$ is the $\delta$-neighborhood of $A$.

Call such a set $A$ a \textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\it extreme point} of $u$.

(b). For $u\in\mathscr{D}_\mu$, denote
\[{N}(u)=\#\{A\subset K: A \text{ is an extreme set of } u\}.\]
For $x>0$, define
\[N(x)=\sup\{{N}(u): u \text { is a $\lambda$-eigenfunction with}\ 0\leq \lambda\leq x\},\]
and call it \textit{the extremum counting function} of $-\Delta_\mu$.
\end{definition}

\noindent{\bf Remark.}
For a $\lambda$-eigenfunction with $\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative).
Indeed, suppose that $u$ has a local maximum at $A$ with value $c\leq 0$. Since $u$ is a $\lambda$-eigenfunction, there exists $\delta>0$ such that $u\leq c\leq 0$ and $\Delta_\mu u=-\lambda u\geq 0$ on $A_\delta$. Moreover, $u$ is non-constant on $A_\delta$. Let $h$ be the harmonic function satisfying $h|_{\partial A_\delta}=u|_{\partial A_\delta}$, and denote by $g(\cdot,\cdot)$ the Green function for $-\Delta_\mu$ on $A_\delta$. It is known that $g(p,q)>0$ for all $p,q\in A_\delta$. Then, for $p\in A$, we have
\begin{equation*}
u(p)=\int_{A_\delta}g(p,q)\lambda u(q)\mathrm{d}\mu(q)+h(p)<h(p)\leq \max u|_{\partial A_{\delta}},
\end{equation*}
which yields a contradiction.

We have certain asymptotic estimates of $N(x)$ as $x\to \infty$.

\noindent{\bf(A)}: there exists $\lambda_0>0$ such that ${N}(u)\leq 1$ for any $\lambda$-eigenfunction $u$ with $0\leq\lambda<\lambda_0$.

\begin{proposition}\label{prothm1}
If condition (A) holds, then
\begin{equation*}
\limsup_{x\to\infty}\frac{N(x)}{x^{d_S/2}}< \infty.
\end{equation*}
\end{proposition}

\begin{proposition}\label{prothm2}
If there exists a pre-localized eigenfunction, then
\begin{equation*}
\liminf_{x\to\infty}\frac{\log N(x)}{\log x}\geq\kappa
\end{equation*}
for some $0<\kappa\leq d_S/2$.
In particular, for the lattice case (see the exact meaning in Section \ref{sec2}),
\begin{equation*}
\liminf_{x\to\infty}\frac{N(x)}{x^{d_S/2}}>0.
\end{equation*}
\end{proposition}

We structure the paper as follows.

In Section \ref{sec2}, we present the proofs of  Propositions \ref{prothm1} and \ref{prothm2}, and provide an equivalent characterization of condition (A).

Beginning in Section \ref{sec3}, we focus on the canonic Laplacian on the Sierpinski gasket $\mathcal{SG}$. There, we recall the spectral decimation method and state two key preparatory theorems — Theorems \ref{thm4} and \ref{thm5}.  The former confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions.

The proof of  Theorem \ref{thm4} is given in Section \ref{sec4},  followed by the proof of Theorem \ref{thm5} in Section \ref{sec5}. We conclude in Section \ref{sec6} with the proof of Theorem \ref{equnthm1}.

The post-theorem introduction says that the proof of Theorem~\ref{thm1} “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.” It further indicates the overall organization supporting Theorem~\ref{thm1}: in Section~\ref{sec3} the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems — Theorems~\ref{thm4} and \ref{thm5},” where “\ref{thm4} confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions,” and then “conclude[s] in Section~\ref{sec6} with the proof” of the main estimate \eqref{equnthm1} (i.e. Theorem~\ref{thm1}).

The post-theorem introduction says that the proof of the main theorem “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \cite{RT} and later rigorously established by Shima and Fukushima \cite{Sh, FS}.” It further indicates the overall organization supporting the main theorem: next the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems.” We first record the following theorem.

\begin{theorem}\label{thm4}
For $0<\lambda<\lambda_1^D$, we have
\begin{equation}\label{eq3}
\begin{aligned}
\mathcal{A}_\lambda=\ & \mathcal{C}_\lambda=\mathcal{D}_{\psi^{-1}(\lambda)},\quad \mathcal{B}_\lambda^1=\big(\bigcup_{ij\in S_1}\mathcal{L}_{\psi^{-1}(\lambda),ij}\big)\cup\{\boldsymbol{\theta}\},\quad\text{and}\\
&(\mathcal{T}_\lambda^i)^{-1}(\mathcal{A}_{5^{-1}\lambda})=
\mathcal{G}_{\psi^{-1}(\lambda),i}\quad\text{for }i\in S.
\end{aligned}
\end{equation}
In particular, condition (A) holds.
\end{theorem}

The introduction then points to the later preparatory result
\begin{theorem}\label{thm5}
For  $\eps\in\{-1,1\}^n$ of length $n\geq 1$ with $\eps_n=1$, we have
\begin{equation}\label{equn}
c^{-1}(\lambda^{\eps})^{d_S/2}\leq N(u^{\eps})\leq c(\lambda^{\eps})^{d_S/2}
\end{equation}
with some constant $c>1$ independent of $\lambda_0$, $\mathbf a$ and $\eps$.
\end{theorem}

and finally says it “conclude[s] later with the proof” of the main estimate
\begin{equation}\label{equnthm1}
C^{-1}\lambda^{d_S/2}\leq{N}(u)\leq C\lambda^{d_S/2}
\end{equation}
(i.e. the main theorem).

Question: On the Sierpiński gasket \(\mathcal{SG}\), let \(N(u)\) denote the number of extreme sets of a function \(u\), where an extreme set is a connected component \(A\) of a level set \(u^{-1}(c)\), disjoint from the boundary vertex set \(V_0\), such that for some \(\delta>0\) the function is everywhere \(\le c\) on the \(\delta\)-neighborhood of \(A\) (local maximum case) or everywhere \(\ge c\) there (local minimum case). Let \(d_S\) be the spectral dimension of \(\mathcal{SG}\). A global Dirichlet or Neumann eigenfunction means an eigenfunction of the Laplacian on all of \(\mathcal{SG}\) satisfying Dirichlet or Neumann boundary conditions on \(V_0\). Which uniform estimate holds for such eigenfunctions?

There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2}\), except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).

Choices:
B. There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2}\), with no exceptional Neumann eigenfunction.
C. There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(N(u)\le C\lambda^{d_S/2}\).
D. For every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), there exists a constant \(C=C(u)>1\) such that \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda^{d_S/2}\), except for the first non-constant Neumann eigenfunction where \(N(u)=0\).
E. There exists a constant \(C>1\) such that for every global Dirichlet or Neumann eigenfunction \(u\) on \(\mathcal{SG}\) with eigenvalue \(\lambda\), one has \(C^{-1}\lambda^{d_S/2}\le N(u)\le C\lambda\), except for the first non-constant Neumann eigenfunction, for which \(N(u)=0\).