\begin{theorem}\label{t:main}
	Let $G$ be an $n$-vertex digraph with $\alpha^*(G) \le s$.
	Then, $f(G) \ge n^{5/9-o(1)}s^{-14/9}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then
	$f(G) \ge n^{5/9-o(1)}$.
\end{theorem}

Throughout this paper, the chromatic number $\chi(G)$, the independence number $\alpha(G)$, and the bipartite independence number $\alpha^*(G)$\,\footnote{Recall that $\alpha(G)$ is the largest $k$ for which there is an empty $k$-vertex set and that $\alpha^*(G)$ is the largest $k$ for which there are two disjoint $k$-vertex sets with no edges between them.} of a digraph $G$ are inherited from the corresponding parameters of its underlying undirected graph.
An {\em orientation} (a.k.a. oriented graph) is a digraph without directed cycles of length $2$. A {\em tournament} is an orientation of a complete graph. A digraph is {\em acyclic} if it has no directed cycles.

Graph colorings and orientations are closely related topics whose origins date back to the
classical theorem of Gallai–Hasse–Roy–Vitaver \cite{gallai-1968,hasse-1965,roy-1967,vitaver-1962}
asserting that every orientation of a $k$-chromatic graph has a directed path with $k$ vertices.
Many variations and related results connecting these concepts can be found in the survey \cite{havet-2013}.
A very general variant of the Gallai–Hasse–Roy–Vitaver Theorem was conjectured by Burr \cite{burr-1980},
stating that any $(2k-2)$-chromatic digraph contains every oriented tree on $k$ vertices.
The special case of Burr's conjecture for tournaments is known as
Sumner's conjecture which has been resolved, for $k$ sufficiently large, by K\"uhn, Mycroft, and Osthus
\cite{KMO-2011}. The best bound on the chromatic number of digraphs which guarantees the containment of every oriented tree on $k$ vertices
is $O(k^{1.5})$ which was recently obtained by Bessy, Gon{\c{c}}alves, and Reinald \cite{BGR-2024}.

Burr's conjecture is known to hold for acyclic digraphs. In fact, it was proved by
Addario-Berry, Havet, Sales, Reed, and Thomass\'e \cite{addario+-2013} that every $k$-chromatic acyclic digraph
contains every oriented tree on $k$ vertices. It would therefore be of interest to prove that
digraphs have acyclic subgraphs that retain the original chromatic number to some nontrivial extent.
Indeed, this problem was formulated in \cite{addario+-2013}.
For a digraph $G$, let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$
and let $f(k)$ be the minimum of $f(G)$ taken over all $k$-chromatic digraphs.
It is a folklore argument (see below) that $f(G) \ge \sqrt{\chi(G)}$ (whence $f(k) \ge \sqrt{k}$), and it was proved in \cite{addario+-2013}
that $f(k) \ge 1+\lfloor \sqrt{k-1} \rfloor$. Trivially, $f(3)=2$ and it is known that $f(4)=3$ \cite{NY-2019,shapira-2022}. The special case of this parameter for the case of tournaments was considered
by Nassar and the author in \cite{NY-2019}. Let $g(k)$ be the maximum integer such that every $k$-vertex tournament has an acyclic subgraph with chromatic number $g(k)$. It was conjectured in \cite{NY-2019}
that $g(k) = \omega(\sqrt{k})$, namely that the folklore lower bound can be improved by more than a constant factor. This conjecture has recently been resolved by Fox, Kwan, and Sudakov \cite{FKS-2021} who proved by an
ingenious argument that $g(k)=\Omega(k^{5/9-o(1)})$. In that same paper, they have shown that the exponent $5/9$
cannot be improved beyond $3/4$. Returning back to $f(k)$, this result suggests that $f(k)=\omega(\sqrt{k})$ as well.
Our main result shows that this is the case in a dense regime.

We shall require the following result of Gallai and Milgram \cite{GM-1960} (see also \cite{diestel-2005}).
\begin{lemma}\label{l:gm}{\rm\cite{GM-1960}}
 Let $G$ be a digraph with $\alpha(G)=s$. Then there is a set of $s$ pairwise vertex-disjoint directed paths of $G$ that cover all its vertices.
\end{lemma}

For the rest of this paper, all digraphs are assumed to be orientations. Clearly this assumption does not change the chromatic number, independence number, or bipartite independence number.
For a subset $S$ of vertices of an oriented graph $G$, let $G[S]$ denote the induced subgraph of $G$ on $S$.
We say that $S$ is an {\em acyclic $k$-set} if $|S|=k$ and $G[S]$ is acyclic.
\begin{lemma}\label{l:prob}
 Let $G$ be a $k$-vertex orientation with $\alpha(G) \le s$. For a random permutation $\pi$, the probability that $G_\pi$ has no edges is at most $(se/k)^k$.
\end{lemma}
\begin{proof}
 By Lemma \ref{l:gm}, $G$ has a set of $s$ pairwise vertex-disjoint paths $P_1,\ldots,P_s$ that cover each vertex.
 Let $p_i$ denote the number of vertices of $P_i$ (possibly $p_i=0$) and observe that $\sum_{i=1}^s p_i = k$.
 Notice that for $G_\pi$ to have no edges, all vertices of $P_i$ must
 appear in the reverse order they appear in $P_i$, which, for a random $\pi$, occurs with probability
 $1/p_i!$. As the $P_i$'s are pairwise vertex-disjoint, we have that the probability that
 $G_\pi$ has no edges is therefore at most the reciprocal of $\prod_{i=1}^s p_i!$.
 Finally note that by logarithmic convexity of $x^x$,
 $$
 \prod_{i=1}^s p_i! \ge \prod_{i=1}^s \left( \frac{p_i}{e}\right)^{p_i} \ge \left( \frac{\sum_{i=1}^s p_i}{se}\right)^{\sum_{i=1}^s p_i}=\left( \frac{k}{se}\right)^{k}\;.
 $$
\end{proof}
\begin{corollary}\label{coro:prob}
 If $G$ is an $n$-vertex orientation with $\alpha(G) \le s$ and $G$ has fewer than $(k/se)^k$ acyclic $k$-sets, then there is a permutation $\pi$ such that $\alpha(G_\pi) \le k$. In particular, $\chi(G_\pi) \ge n/k$.
\end{corollary}
\begin{proof}
 Consider a random $\pi$. For a set $S$ of $k$ vertices, the
 probability that $S$ is an independent set in $G_\pi$ is either $0$ if $S$ is not acyclic, or, by Lemma \ref{l:prob}, at
 most $(se/k)^k$ if $S$ is acyclic. Hence, if $G$ has fewer than $(k/se)^k$ acyclic $k$-sets, then by the union bound, $\alpha(G_\pi) \le k$ with positive probability.
\end{proof}

\begin{lemma}\label{l:deg}
 Let $G$ be an $m$-vertex orientation with $\alpha(G) \le s$. Then $G$ has a vertex whose in-degree and out-degree are both at least $m/4s-1/2$.
\end{lemma}
\begin{proof}
 Let $t=m/4s-1/2$. Assume the claim is false and partition $V(G)$ into parts $V_1$ and $V_2$ where all vertices of $V_1$ have out-degree smaller than $t$ and all vertices of $V_2$ have in-degree smaller than $t$. Without loss of generality, $|V_1| \ge m/2$. Consider the underlying undirected graph of $G[V_1]$. Since $\alpha(G[V_1]) \le s$, there is a subgraph of this undirected graph
 with minimum degree at least $|V_1|/s-1 \ge m/2s-1$. Returning to the directed version, there must be some vertex in this subgraph whose out-degree is not smaller than its in-degree, hence its out-degree is at least $(m/2s-1)/2=t$, contradicting the definition of $V_1$.
\end{proof}

\begin{lemma}\label{l:l22}
 Let $k = n^{\Omega(1)}$ and $k \ge s^7$. If an $n$-vertex orientation $G$ with $\alpha(G) < s$ and $\alpha^*(G) < s$ has at least $(k/se)^k$ acyclic $k$-sets, then it has a set of $n' \ge k^{2-o(1)}/s$ vertices inducing a $q$-almost-acyclic subgraph $G'$
 where $q = n's^6/k^{1-o(1)}$.
\end{lemma}

\begin{lemma}\label{l:l23}
Suppose $n^{1/3}s^{-2/3} \le q \le n^{1-\Omega(1)}$. An $n$-vertex $q$-almost-acyclic orientation $G$ with
$\alpha(G) < s$ has a permutation $\pi$ with $\alpha(G_\pi) \le n^{1/4+o(1)}q^{1/4}s^{3/2}$, so $\chi(G_\pi) \ge n^{3/4-o(1)}q^{-1/4}s^{-3/2}$.
\end{lemma}

\begin{lemma}\label{l:l24}
 Let $n$ be sufficiently large and let $G$ be an $n$-vertex orientation with $\alpha(G) < s$ and $\alpha^*(G) < s$ having at least
 $M$ acyclic $k$-sets. If $q = 6000s^6 n\log^2n/k$, then there is $k'$ satisfying $k-3k/\log n \le k' \le k$ such that $G$ has an induced $(q,6sk/\log n)$-almost-acyclic subgraph $G'$ on at least $e^{-8}kM^{1/k}$ vertices having at least $e^{-6k}M$ acyclic $k'$-sets.
\end{lemma}
\begin{proof}
 Let $t$ be a positive integer parameter to be set later.
 For each $i=0,\ldots,t$ we define vertex sets $W_i,V_i,V'_i$ such that
 \begin{enumerate}
 \item
 $W_i$ is an ordered acyclic $i$-set, $W_i \subset W_{i+1}$, and the ordering of $W_{i+1}$ is consistent with the ordering of $W_i$.
 \item
 $V_0'=V(G)$, $V_i' \supseteq V_i \supset V_{i+1}' \supseteq V_{i+1}$ for $i=0,\ldots,t-1$.
 \item
 $V_i' \cap W_i = \emptyset$.
 \end{enumerate}
 Describing the iterative procedure building these sets requires several notions.
 \begin{itemize}
 \item
 Let $H'_i$ be the $(k-i)$-uniform hypergraph with vertex set $V'_i$ whose edges are the
 $(k-i)$-subsets $S$ for which $G[S \cup W_i]$
 is an acyclic $k$-set for which there is a topological sort $\sigma_S$ consistent with the ordering of $W_i$.
 Let $N'_i$ denote the number of edges of $H'_i$ and observe that $N'_0 \ge M$.
 \item
 Let $H_i$ be an induced subhypergraph of $H'_i$ with minimum degree at least $N'_i/|V'_i|$ (it will always be the case that $N'_i > 0$). Such a subhypergraph can be obtained by repeatedly removing vertices with degree smaller than
 $N'_i/|V'_i|$ until none are left. Let $N_i$ be the number of edges of $H_i$
 and let $V_i$ be its vertex set.
 \item
 Let the ordering of $W_i$ be $\{w_1,\ldots,w_i\}$ so that if $(w_j,w_{j'}) \in E(G)$, then $j < j'$.
 Consider some vertex $v \in V$ and some edge $S$ of $H_i$ containing $v$.
 Recall that $G[S \cup W_i]$ is an acyclic $k$-set for which $\sigma_S$ is a topological sort consistent with the ordering of $W_i$. So if $(w_j,v) \in E(G)$ and $(v, w_{j'}) \in E(G)$, it must be that $j < j'$. Thus, all out-neighbors of $v$ in $W_i$ (if there are any) appear in $W_i$'s ordering after all in-neighbors of $v$ in $W_i$ (if there are any). We say that the set of {\em allowed bins} of $v$ with respect to $W_i$ is $\{j,\ldots,j'\}$
 if $(w_j,v) \in E(G)$, $(v,w_{j'+1}) \in E(G)$ and $v$ is not adjacent in $G$ to any of
 $w_{j+1},\ldots,w_{j'}$. If $v$ has no in-neighbors in $W_i$, then let $j=0$ and if $v$ has no out-neighbors in $W_i$, then let $j'=i$. Notice that for any edge $S$ of $H_i$ containing $v$,
 $\sigma_S$ places $v$ after some $w_\ell$ and before some $w_{\ell+1}$ where $j \le \ell \le j'$ (if $\ell = 0$, then $j=0$ so $v$ is placed before $w_1$ and if $\ell =i$, then $j'=i$ so $v$ is placed after all $w_i$). We say that bin $\ell$ is {\em associated with $(\sigma_S,v)$}. One may think of $\{w_1,\ldots,w_i\}$ as labeling the consecutive internal walls of a set of $i+1$ bins labeled $0,\ldots,i$.
 \item
 Two vertices $x,y$ of $V_i$ are {\em incompatible} if $(x,y) \in E(G)$ and every allowed bin of $x$ with respect to $W_i$ is {\em after} every allowed bin of $y$ with respect to $W_i$
 (in particular, no edge of $H_i$ contains both $x$ and $y$).
 \end{itemize}
 For $0 \le i \le t-1$, we shall go from step $i$ to step $i+1$ by picking some (carefully selected) vertex $w \in V_i$ and some (carefully selected) allowed bin $\ell$ of $w$. We add $v$ to $W_i$ to obtain $W_{i+1}$ with the ordering of $W_{i+1}$ being $\{w_1,\ldots,w_\ell,w,w_{\ell+1},\ldots,w_i\}$.
 We remove from $V_i$ all vertices incompatible with $w$ to obtain $V'_{i+1}$. The edges of $H'_{i+1}$ are obtained by taking all edges $S$ of $H_i$ which contain $w$ and for which $(\sigma_S,w)$ is associated with $\ell$, and keeping $S \setminus w$ as an edge of $H'_{i+1}$.

\noindent
Theorem \ref{t:main} extends the result of Fox, Kwan, and Sudakov which is the case $s=0$.
Notice that if $\alpha^*(G) \le s$, then $\alpha(G) \le 2s+1$ and thus $\chi(G) \ge n/(2s+1)$.
So, Theorem \ref{t:main} significantly improves upon the folklore $\sqrt{\chi(G)}$ lower bound
for graphs with relatively small bipartite independence number.
Theorem \ref{t:main} is particularly interesting in the setting of {\em random graphs}.
Let ${\cal G}(n,p)$ denote the  Erd\H{o}s–R\'enyi random $n$-vertex graph with edge probability $p$.
It is well known that if $p = n^{-o(1)}$, then $G \sim {\cal G}(n,p)$ almost surely\footnote{As usual, ``almost surely'' means with probability tending to $1$ as $n$ tends to infinity.} has
$\chi(G) =n^{1-o(1)}$ and $\alpha^*(G)=n^{o(1)}$ (see \cite{bollobas-1988} for sharp estimates when $p$ is constant). We therefore have:
\begin{corollary}\label{coro:main}
	Let $p = n^{-o(1)}$ and $G \sim {\cal G}(n,p)$. Then almost surely,
	every orientation $\vec{G}$ of $G$ has $f(\vec{G}) \ge n^{5/9-o(1)}$. Using $p=1/2$ we have that almost all $n$-vertex graphs $G$ have the property that every orientation $\vec{G}$ of $G$ has $f(\vec{G}) \ge n^{5/9-o(1)}$.
\end{corollary}

To prove Theorem \ref{t:main}, we first use a theorem of Gallai and Milgram on the path-covering number of digraphs with given independence number. This is then used, together with some additional ideas, to effectively
reduce the proof to the same framework of the proof of Fox, Kwan, and Sudakov. We do, however, need to (sometimes extensively) modify their arguments in locations where their proof relies on the fact that every pair of vertices is adjacent, which is not the case in our setting. In Section \ref{sec:lem} we prove several lemmas needed for our proof.
In Section \ref{sec:fks} we revisit the proof of Fox, Kwan and Sudakov, state variants of their
Lemmas 2.2 (which is Lemma \ref{l:l22} here) and 2.3 (which is Lemma \ref{l:l23} here) and prove
Lemma \ref{l:l23}. Lemma \ref{l:l22}, which is the main technical lemma, is proved in Section \ref{sec:l22}. Throughout the paper, we ignore rounding issues as long as they do not affect the asymptotic claims.

We end the introduction recalling the aforementioned folklore lower bound. Clearly, in every red/blue edge coloring
of a graph $G$, $\chi(G_r) \cdot \chi(G_b) \ge \chi(G)$ where $G_r$ and $G_b$ respectively denote the red and blue subgraphs. For every digraph $G$ and every linear order of its vertices, we can color red the edges going from left to right and color blue the edges going from right to left, observing that the obtained $G_r$ and $G_b$ are acyclic. Hence $\max(\chi(G_r),\chi(G_b)) \ge \sqrt{\chi(G)}$.
Nevertheless, despite the similarity between acyclic orientations and red/blue edge colorings, note that an analogous result such as Theorem \ref{t:main} is, in general, false for red/blue edge colorings. To see this, consider the complete $n/s$-partite
graph with each part of order $s$; clearly in this case $\alpha^*(G) \le \alpha(G)=s$.
Further partition the parts into $\sqrt{n/s}$ buckets, each bucket consisting of$\sqrt{n/s}$ parts. Now color blue all the edges with endpoints in parts belonging to the same bucket
and color red all remaining edges. It is immediate to verify that $G_r$ and $G_b$ are both $\sqrt{n/s}$ chromatic.

To prove the main theorem, we first use a theorem of Gallai and Milgram on the path-covering number of digraphs with given independence number. This is then used, together with some additional ideas, to effectively reduce the proof to the same framework of the proof of Fox, Kwan, and Sudakov. We do, however, need to (sometimes extensively) modify their arguments in locations where their proof relies on the fact that every pair of vertices is adjacent, which is not the case in our setting. Next we prove several lemmas needed for our proof. We then revisit the proof of Fox, Kwan and Sudakov, state variants of their Lemmas 2.2 (which is
\begin{lemma}\label{l:l22}
	Let $k = n^{\Omega(1)}$ and $k \ge s^7$. If an $n$-vertex orientation $G$ with $\alpha(G) < s$ and $\alpha^*(G) < s$ has at least $(k/se)^k$ acyclic $k$-sets, then it has a set of $n' \ge k^{2-o(1)}/s$ vertices inducing a $q$-almost-acyclic subgraph $G'$
	where $q =  n's^6/k^{1-o(1)}$.
\end{lemma}
here) and 2.3 (which is
\begin{lemma}\label{l:l23}
Suppose $n^{1/3}s^{-2/3} \le q \le n^{1-\Omega(1)}$. An $n$-vertex $q$-almost-acyclic orientation $G$ with
$\alpha(G) < s$ has a permutation $\pi$ with $\alpha(G_\pi) \le n^{1/4+o(1)}q^{1/4}s^{3/2}$, so $\chi(G_\pi) \ge n^{3/4-o(1)}q^{-1/4}s^{-3/2}$.
\end{lemma}
here) and prove the lemma above. The lemma above, which is the main technical lemma, is proved later.

Question: Let $G$ be an $n$-vertex orientation (that is, a digraph with no directed $2$-cycle). The bipartite independence number $\alpha^*(G)$ is the largest integer $k$ for which there exist two disjoint $k$-vertex sets with no edges between them. Let $f(G)$ denote the maximum chromatic number of an acyclic subgraph of $G$, where chromatic number is taken for the underlying undirected graph and acyclic means having no directed cycle. If $\alpha^*(G) \le s$, which quantitative lower bound holds for $f(G)$, and what special case follows when $\alpha^*(G)=n^{o(1)}$?

Choices:
B. $f(G) \ge n^{5/9-o(1)}s^{-3/2}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$.
C. $f(G) \ge n^{1/2-o(1)}s^{-14/9}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{1/2-o(1)}$.
D. $f(G) \ge n^{5/9-o(1)}s^{-14/9}$ whenever $\alpha(G) \le s$. In particular, if $\alpha(G)=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$.
E. $f(G) \ge n^{3/4-o(1)}s^{-3/2}$. In particular, if $\alpha^*(G)=n^{o(1)}$, then $f(G) \ge n^{3/4-o(1)}$.

final s-exponent from combining technical lemmas

hypothesis uses bipartite independence number rather than independence number

imports intermediate 3/4 exponent from almost-acyclic lemma as final theorem

The stem does not reveal the correct option directly; it asks for a quantitative bound, so the answer is not leaked from the wording alone.

This is essentially a direct restatement of a specific theorem/result, asking the test-taker to identify the exact bound rather than derive or compare substantive conclusions.

Some discrimination is required among close exponents and parameter variants, but the task is mainly recall/recognition of the theorem statement rather than genuine mathematical generation or derivation.

The distractors are mathematically plausible: they vary exponents, swap in a weaker true bound, confuse independence number with bipartite independence number, or promote an intermediate estimate to the final claim.

A solid theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than generative understanding.


\begin{cor}
\label{cor:cycles}
The graph $\Gkp$ contains cycles
of lengths $4n+2$, $6n+3$, and $8n+2$ if there exists $n\in\mathbb{Z}_{>0}$ such that $\Knkpprop\ne\emptyset$.
\end{cor}

\begin{cor}
\label{cor:sufficient condition non-planarity}
The graph $\Gkp$ is non-planar if there exists $n\in\mathbb{Z}_{>0}$ such that $\Knkpprop\ne\emptyset$.
\qed
\end{cor}

\begin{theorem}
\label{thm:main}
Let $\kappa \in\mathbb{Z}\setminus \{4\}$.
Then the followings hold:
\begin{enumerate}
\item[$(1)$]
$(\mathrm{Theorems}~\ref{thm-n=1},\ \ref{thm:n=2 non-planarity} \  \text{and}\ \ref{thm-n=(p-1)/2})$
For each such $\kappa$ and for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/2$,
there exists an explicit $K_{3,3}$-subdivision in $\Gkp$.

\item[$(2)$]
$(\mathrm{Theorem}~\ref{thm:np density})$
Moreover, for almost all $\kappa$,
the natural density of primes $p$ for which $(1)$ holds is at least $13/16$.

\end{enumerate}
\end{theorem}

\begin{theorem}
\label{thm-proj}
Let $p>3$ be a prime. Then we have the followings.
\begin{enumerate}
\item[$(1)$] The graph $\Gzp$ is toroidal if and only if $p=7$.
\item[$(2)$] The graph $\Gzp$ is projective-planar if and only if $p=7$.
\end{enumerate}
\end{theorem}

The Diophantine equation
\begin{equation}
\label{def:Markoff-equation}
x^2+y^2+z^2=xyz
\end{equation}
is called the {\it Markoff equation},
and a solution $(x,y,z)$ is called a {\it Markoff triple}.
For a prime $p>3$,
let $\Mzp$ denote the set of all Markoff triples over $\Fp$, the finite field with $p$ elements.
The {\it Markoff mod $p$ graph} $\Gzp$ is a finite $3$-regular graph (possibly with loops) whose vertex set is $\Mzp \setminus \{(0,0,0)\}$, and two triples are adjacent if and only if they are transformed into each other by one of the following {\it Vieta involutions}:
\[
R_1(x,y,z)=(yz-x,y,z), \quad
R_2(x,y,z)=(x,zx-y,z), \quad
R_3(x,y,z)=(x,y,xy-z).
\]

The graph $\Gzp$ has been extensively studied in the literature on ``strong approximation'' for the Markoff equation (\ref{def:Markoff-equation}).
Indeed, Bourgain, Gamburd, and Sarnak~\cite{BGS2016} conjectured that $\Gzp$ is connected for all primes $p>3$.
Recently Chen~\cite{C2024} made a breakthrough toward this conjecture by proving that $\Gzp$ is connected for any sufficiently large prime $p$.
A key result here is that the size of any component of $\Gzp$ must be divisible by $p$. Furthermore, for any $\varepsilon>0$, there are only $O(p^{\varepsilon})$ vertices that are not contained in the giant component (\cite{BGS2016}) while $\Gzp$ has approximately $p^2$ vertices (see (\ref{eq-number}) below).
Martin~\cite{Martin2025} provided an elementary proof of Chen's result concerning the divisibility of the component sizes of $\Gzp$.
In addition, Eddy, Fuchs, Litman, Martin, and Tripeny~\cite{EFLMT2025} established an explicit lower bound for primes $p$ for which the connectivity of $\Gzp$ is guaranteed, namely, $p>3.448\times10^{392}$.
There are several algorithmic results towards the connectivity of $\Gzp$, and it was verified in \cite{B2025} that $\Gzp$ is connected for $p<10^6$.

On the other hand, de Courcy-Ireland~\cite{C2024} established the non-planarity of $\Gzp$: $\Gzp$ is planar if and only if $p=7$. This provides (indirect but positive) evidence for another well-known conjecture by Bourgain, Gamburd, and Sarnak~\cite{BGS2016, BGS2026} asserting that $\{\Gzp\}_{\text{$p$:prime}}$ is in fact an expander family, which would imply higher connectivity for $\Gzp$ (for further details on expander family, see e.g. \cite{HLW2004, L2012}).
The proof of non-planarity consists of two parts: an asymptotic evaluation of the Euler characteristic
of $\Gzp$ and explicit constructions of subdivisions of the complete bipartite graph $K_{3,3}$.
The Kuratowski-Wagner theorem shows a complete characterization for planar graphs in terms of graph minors, that is, a graph is planar if and only if it contains neither $K_{3,3}$ nor the complete graph $K_5$ as a graph minor.
Since $\Gzp$ is a $3$-regular graph, $K_{3,3}$ is the unique obstruction to planarity of $\Gzp$ in the sense that
a $3$-regular graph cannot contain a subdivision of $K_5$, and more generally, any $3$-regular graph containing $K_5$ as a minor must also contain a subdivision of $K_{3,3}$.

There are several generalizations of \eqref{def:Markoff-equation}, such as the following equation: for a (rational) integer $\kappa$,
\begin{equation}
\label{def:Markoff-type equation}
x^2+y^2+z^2=xyz+\kappa.
\end{equation}
Throughout this paper, we naturally treat $\kappa$ as an element of $\Fp$ when considering the solutions of \eqref{def:Markoff-type equation} modulo $p$.
For the set of all such solutions over $\Fp$, denoted by $\Mkp$,
Carlitz \cite{C1954} proved that
\begin{align}
\label{eq-number}
 \# \Mkp
=p^2+\left(3+\left(\frac{\kappa}{p}\right)\right)\left(\frac{\kappa-4}{p}\right)p+1,
\end{align}
where $(\frac{\cdot}{p})$ is the Legendre symbol.
The {\it generalized Markoff mod $p$ graph} $\Gkp$ (with respect to \eqref{def:Markoff-type equation}) is defined as a $3$-regular graph
whose vertex set is $\Mkp$
with the adjacency relation defined in the same manner as for $\Gzp$ (hence $\mathcal{G}_0(p)$ coincides with $\Gzp \sqcup \{(0,0,0)\}$, where $(0,0,0)$ is an isolated vertex).
The graph structure of $\Gkp$ depends sensitively on $\kappa$ and $p$ in general.
For example, in the case of $\kappa=4$ (corresponding to a degenerate surface called the {\it Cayley cubic}, cf. \cite{CL2020}), the graph $\mathcal{G}_4(p)$ possesses many small components.
While $\Gkp$ is no longer connected in general,
it is still expected that for $\kappa\neq 4$,
there exists a unique {\it giant component} $\mathcal{C}_\kappa(p)$ in
$\Gkp$ such that $\#(\Gkp \setminus \mathcal{C}_\kappa(p))=o_\kappa(p^2)$. In a recent preprint~\cite{Mart2025}, Martin announced that this in fact holds for a majority of primes.
Moreover, the main result of \cite{Mart2025} would imply that $\#(\Gkp \setminus \mathcal{C}_\kappa(p))=O(1)$, representing a significant advance in the understanding of the global structure of $\Gkp$.
However, the precise structure of $\Gkp$ (and $\mathcal{C}_\kappa(p)$) is still largely unexplored.

In this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.
In particular, by leveraging computational algebraic methods including resultants and Gr\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.
Throughout this paper, the term {\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.

In this paper, we investigate the topological properties of the generalized Markoff mod $p$ graph $\Gkp$ by combining perspectives of graph theory, computer algebra, and number theory.
In particular, by leveraging computational algebraic methods including resultants and Gr\"obner bases, in conjunction with the Chebotarev density theorem, we derive the following results based on graph minor theory.
Throughout this paper, the term {\it $K_{3,3}$-subdivision} refers to a subdivision of the complete bipartite graph $K_{3,3}$.

Theorem~\ref{thm:main} yields several topological properties of $\Gkp$.
In particular, the following corollary is an immediate consequence of Theorem~\ref{thm:main}.

\begin{theorem}[Theorem~\ref{thm-disjoint-k}]
\label{thm:disjoint-intro}
For each $\kappa \in \mathbb{Z} \setminus \{ 4\}$, $\Gkp$ contains at least four mutually vertex-disjoint $K_{3,3}$-subdivisions for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/4$ if $\kappa=2$ and $1/2$ otherwise.
Moreover, for almost all $\kappa$, the natural density of primes $p$ for which this holds is at least $5/8$.
\end{theorem}

\begin{theorem}
\label{thm:n=2 non-planarity}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$ if $\xi_{2,\kappa}\eta_{2,\kappa}\ne 0$, $5(\kappa^2-73\kappa+61)\ne 0$,
and either of the following conditions holds.
\begin{itemize}
\item[$\mathrm{(A)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=1$ and
$\left(\dfrac{5}{p}\right)
=\left(\dfrac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)
=\left(\dfrac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1$, or,
\\[0pt]
\item[$\mathrm{(B)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=-1$ and $\left(\dfrac{5}{p}\right)=1$.
\end{itemize}
\end{theorem}

\subsection{Natural densities}
\label{subsec:density}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
From Theorem~\ref{thm-n=1}, Theorem~\ref{thm:n=2 non-planarity} (B),
or Theorem~\ref{thm-n=(p-1)/2},
it follows from Dirichlet's theorem on arithmetic progressions that there are infinitely many primes $p$ for which
$\Gkp$ contains a $K_{3,3}$-subdivision, which yields topological properties of $\Gkp$ such as the non-planarity and existence of short cycles by Corollaries~\ref{cor:sufficient condition non-planarity} and \ref{cor:cycles}, respectively.
This motivates us to consider the natural density of such primes $p$: \[
\delta_{\kappa}
\coloneq\lim_{x\to\infty}
\frac{\#\{p\le x\,|\,\text{$\Gkp$ contains a $K_{3,3}$-subdivision}\}}{\#\{p\le x\}}.
\]
It holds that $\delta_0=1$
since $\mathcal{G}(p)$ is non-planar for any $p\neq 7$
by \cite[p.115]{C2024}.
For any $\kappa \neq 4$, Theorem~\ref{thm-n=1} and the Chebotarev density theorem (see, e.g.~\cite{N1999})
imply that $ \delta_{\kappa}\geq 1/2$.
Moreover, combining the three theorems above,
we arrive at the following theorem.

\begin{theorem}
\label{thm:np density}
\begin{itemize}
\item[$(1)$]
For each $\kappa\in\mathbb{Z}\setminus\{4\}$ such that none of
$\eta_{1,\kappa}$,
$\eta_{2,\kappa}$,
$5$, and $\kappa-4$
is a square in $\mathbb{Z}$,
and that the ratio of any two of them is not a square in $\mathbb{Q}$,
we have
\begin{equation}
\label{for:lower bound of Dnp}
\delta_{\kappa}\ge \frac{13}{16}.
\end{equation}
That is, $\Gkp$ contains a $K_{3,3}$-subdivision
for at least $81.25\%$
of primes $p$ asymptotically.
\item[$(2)$]
The natural density
of $\kappa\in\mathbb{Z}\setminus\{4\}$ for which $(1)$ holds
is equal to $1$.
\end{itemize}
\end{theorem}

\begin{theorem}
\label{thm-disjoint-k}
\begin{enumerate}
\item[$(1)$]
For each $\kappa\in\mathbb{Z}\setminus\{4\}$,
$\Gkp$ contains at least four mutually vertex-disjoint
$K_{3,3}$-subdivisions for infinitely many primes $p$ of natural density at least $1/4$ if $\kappa=2$ and $1/2$ otherwise.
Furthermore, if $\kappa$ is an integer such that
none of $\eta_{1, \kappa}$, $\eta_{2,\kappa}$ and $5$ is a square in $\mathbb{Z}$ and that the ratio of any two of them is also not a square in $\mathbb{Q}$,
then the natural density of such primes is at least $5/8$.
\item[$(2)$]
The natural density of $\kappa\in\mathbb{Z}\setminus\{4\}$
for which the latter assertion of $(1)$ holds is equal to $1$.
\end{enumerate}
\end{theorem}

\begin{remark}
The four mutually vertex-disjoint $K_{3,3}$-subdivisions above are not necessarily contained in the giant component $\mathcal{C}_{\kappa}(p)$ in general.
Indeed, for $\kappa=2$ and $3$, then
the $K_{3,3}$-subdivision $K\in \mathcal{K}^{\mathrm{prop}}_{1,\kappa}(p)$
obtained from Theorem~\ref{thm-n=1} (along with its three images under double sign changes) is contained in $\mathcal{S}_{2}(p)$ and $\mathcal{S}_{3}(p)$, respectively, where $\mathcal{S}_{2}(p)$ is a unique $16$-vertex component in Remark~\ref{rem-k=2} and $\mathcal{S}_{3}(p)$ denotes a unique $72$-vertex component in $\mathcal{G}_3(p)$ corresponds to the orbit of $((1+\sqrt{5})/2, 1, 1) \in \mathcal{M}_{3}(p)$
under the group generated by Vieta involutions,
provided that $(\frac{5}{p})=1$
(see Figure~\ref{fig:k3smallcomponent}).
On the other hand, according to a recent preprint \cite{Mart2025}, for $\kappa\in \mathbb{Z} \setminus \{2, 3, 4\}$,
all the constructed $K_{3,3}$-subdivisions are expected to be contained within $\mathcal{C}_\kappa(p)$ of $\Gkp$.
Furthermore, even in the cases $\kappa=2$ and $3$,
the $K_{3,3}$-subdivision with 24 vertices constructed in Theorem~\ref{thm:n=2 non-planarity} $(n=2)$ must be contained in $\mathcal{C}_\kappa(p)$.
This follows from the proof of Theorem~\ref{thm-disjoint-k}, together with a simple observation that for $\kappa=2$ and $3$,
any component other than $\mathcal{C}_\kappa(p)$ has at most $72$ vertices
and hence lacks sufficient space to accommodate all four mutually vertex-disjoint copies of the subdivision.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.9\linewidth, height=0.2
\textheight, keepaspectratio]{Small_component_k=3_colored.pdf}
\caption{The $72$-vertex component $\mathcal{S}_3(p)$ of $\mathcal{G}_3(p)$ containing four mutually vertex-disjoint $K_{3,3}$-subdivisions as established in Theorem~\ref{thm-disjoint-k}}
\label{fig:k3smallcomponent}
\end{center}
\end{figure}
\end{remark}

Theorem~\ref{thm:main} yields several topological properties of $\Gkp$.
In particular, the following corollary is an immediate consequence of Theorem~\ref{thm:main}.

\begin{cor}
\label{cor:direct}
Under the same assumptions on $\kappa$ and $p$ of Theorem~\ref{thm:main} $(1)$,
the followings hold:
\begin{enumerate}
\item[$(1)$]
$(\mathrm{Corollary}~\ref{cor:sufficient condition non-planarity})$
$\Gkp$ is non-planar.
\item[$(2)$]
$(\mathrm{Corollary}~\ref{cor:cycles})$
$\Gkp$ has cycles (i.e. closed paths with no repeated vertices or edges) of lengths $6,9,10,15$, and $18$,
all of which can be constructed explicitly.
\end{enumerate}
\end{cor}
Note that it would be interesting to obtain results analogous to Theorem~\ref{thm:main} and Corollary~\ref{cor:direct} for other generalizations of the Markoff mod $p$ graph $\Gzp$ (e.g. \cite{CLM2025}); these will be discussed in our subsequent works.

Recall that $\Gkp$ admits the following non-trivial graph automorphisms, known as {\it double sign changes}:
\[
(x,y,z) \mapsto (x,-y,-z), \quad
(x,y,z) \mapsto (-x,y,-z), \quad \text{and} \quad
(x,y,z) \mapsto (-x,-y,z).
\]
These automorphisms, together with the identity, form a group isomorphic to $(\mathbb{Z}/2\mathbb{Z})^2$.
Combining this symmetry of $\Gkp$
with Theorem~\ref{thm:main},
we obtain an explicit construction of mutually vertex-disjoint $K_{3,3}$-subdivisions.

\begin{theorem}[Theorem~\ref{thm-disjoint-k}]
\label{thm:disjoint-intro}
For each $\kappa \in \mathbb{Z} \setminus \{ 4\}$, $\Gkp$ contains at least four mutually vertex-disjoint $K_{3,3}$-subdivisions for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/4$ if $\kappa=2$ and $1/2$ otherwise.
Moreover, for almost all $\kappa$, the natural density of primes $p$ for which this holds is at least $5/8$.
\end{theorem}

Theorem~\ref{thm:disjoint-intro} strengthens Corollary~\ref{cor:direct} (1). Consider an embedding of $\Gkp$ into a closed surface (i.e. a compact topological surface without boundary); see e.g.~\cite{GT1987, MT2001} for the details on surface embeddings of graphs.
Recall that the Euler characteristic $\chi$ of $\Gkp$ is defined by $V-E+F$, where $V, E$, and $F$
denote the numbers of vertices, edges and faces, respectively.
Combining \cite[Theorem~1.1]{Mart2025} with Theorem~\ref{thm:disjoint-intro},
it follows that $\chi\leq -6$
for $\kappa$ and $p$ as in Theorem~\ref{thm:disjoint-intro},
whereas Euler's formula requires  $\chi=2$ if $\Gkp$ is planar.
Moreover, the existence of mutually vertex-disjoint $K_{3,3}$-subdivisions highlights the robustness of the non-planarity of $\Gkp$ in the sense that, by Theorem~\ref{thm:disjoint-intro}, one must remove at least four vertices to make the graph planar.
It is worth noting that, in light of \cite[Theorem~1.1]{Mart2025}, our construction of $K_{3,3}$-subdivisions implies that the Euler characteristic of the giant component $\mathcal{C}_\kappa(p)$ is at most $-6$ for any $\kappa\neq 4$
and infinitely many primes $p$ (see Section~\ref{sect:disjoint} for details).
From these structural perspectives, our results serve as heuristic evidence for the expander property of the giant component $\mathcal{C}_\kappa(p)$.
At least, $\mathcal{C}_\kappa(p)$ in fact possesses certain complex structure similar to that of $\Gzp$.

It is also worth mentioning that in cryptography and number theory, studying the structure of $\Gkp$ is crucial for understanding the security of the associated cryptographic hash functions proposed in \cite{FLLT2021, Sil2025}, which are expected to be expander hash functions~\cite{CLG2009}.
In particular, it is a challenging open problem to investigate the existence and distribution of short cycles in $\Gkp$, as these properties directly affect the collision resistance of the hash functions (\cite{D2023, FLLT2021}).
Corollary~\ref{cor:direct} (2) provides a modest step toward addressing this problem.

In addition, for $\kappa=0$,
we give a complete characterization for primes $p$ for which the graph $\Gzp$ admits an embedding into closed surfaces of minimal topological complexity beyond the plane.
\begin{theorem}[Theorem~\ref{thm-proj}]
\label{thm:embedded-intro}
Let $p>3$ be a prime. Then the graph $\Gzp$ admits an embedding into the torus
(i.e. the orientable closed surface of Euler characteristic zero)
or the (real) projective plane
(i.e. the non-orientable closed surface of Euler characteristic one)
if and only if $p=7$.
\end{theorem}

This result is achieved through Theorem~\ref{thm:main} by finding suitable forbidden minors for these embeddings, as well as the numerical verification of the connectivity of $\Gzp$ for $p<10^6$ (\cite{B2025}).
Note that the graph $\mathcal{G}(7)$ is planar, and hence one can draw it on any closed surface without edge crossings.
Consequently, Theorem~\ref{thm:embedded-intro} immediately implies the non-planarity of $\Gzp$ for any $p\neq 7$, and hence strengthens the main result of \cite{C2024}.

The rest of this paper is organized as follows.
Section~\ref{sect:chebyshev} introduces the basic and relevant properties of  Chebyshev polynomials, which play a key role in constructing $K_{3, 3}$-subdivisions in $\Gkp$.
In Section~\ref{sec:Six triples}, we present a systematic construction of $K_{3,3}$-subdivisions based on careful computations of Chebyshev polynomials and their associated resultants.
Briefly, we first reduce the construction to the problem of solving systems of algebraic equations via Gr\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.
Then we establish sufficient conditions for the ``properness'', which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.
Based on these arguments, Section~\ref{sect:non-planarity} provides explicit $K_{3,3}$-subdivisions.
Finally, Sections~\ref{sect:disjoint} and \ref{sect:embedded} are devoted to the proofs of Theorems~\ref{thm:disjoint-intro} and \ref{thm:embedded-intro}, respectively.

The introduction outlines the overall strategy leading to Theorem~\ref{thm:main} as follows. Chebyshev polynomials are introduced as a key tool: they “play a key role in constructing $K_{3, 3}$-subdivisions in $\Gkp$.” The construction is then made systematic via “careful computations of Chebyshev polynomials and their associated resultants.” More specifically, “we first reduce the construction to the problem of solving systems of algebraic equations via Gr\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.” After producing these candidates, the next step is to “establish sufficient conditions for the \`\`properness\'\', which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.” Using these arguments, the paper then “provides explicit $K_{3,3}$-subdivisions,” yielding Theorem~\ref{thm:main}.

The introduction outlines the overall strategy leading to the main theorem as follows. Chebyshev polynomials are introduced as a key tool: they “play a key role in constructing $K_{3, 3}$-subdivisions in $\Gkp$.” The construction is then made systematic via “careful computations of Chebyshev polynomials and their associated resultants.” More specifically, “we first reduce the construction to the problem of solving systems of algebraic equations via Gr\"obner bases to obtain candidates for the sextuples of triples that form the desired subdivisions.” After producing these candidates, the next step is to “establish sufficient conditions for the \`\`properness\'\'”, which guarantees the claimed topological properties, of the subdivisions corresponding to the obtained sextuples.” Using these arguments, the paper then “provides explicit $K_{3,3}$-subdivisions,” thereby establishing the main theorem.

\label{thm:main}
Let $\kappa \in\mathbb{Z}\setminus \{4\}$.
Then the followings hold:
\begin{enumerate}
\item[$(1)$]
The following three theorems provide sufficient conditions for the existence of an explicit $K_{3,3}$-subdivision in $\Gkp$:
\begin{theorem}
\label{thm-n=1}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$
if $\xi_{1,\kappa}\eta_{1,\kappa}\ne 0$ and $\left(\frac{\eta_{1,\kappa}}{p}\right)=1$.
\end{theorem}
\begin{theorem}
\label{thm:n=2 non-planarity}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$ if $\xi_{2,\kappa}\eta_{2,\kappa}\ne 0$, $5(\kappa^2-73\kappa+61)\ne 0$,
and either of the following conditions holds.
\begin{itemize}
\item[$\mathrm{(A)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=1$ and
$\left(\dfrac{5}{p}\right)
=\left(\dfrac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)
=\left(\dfrac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1$, or,
\\[0pt]
\item[$\mathrm{(B)}$]
$\left(\dfrac{\eta_{2,\kappa}}{p}\right)=-1$ and $\left(\dfrac{5}{p}\right)=1$.
\end{itemize}
\end{theorem}
\begin{theorem}
\label{thm-n=(p-1)/2}
Let $\kappa\in\mathbb{Z}\setminus\{4\}$.
Then, there exists an explicit $K_{3,3}$-subdivision in $\Gkp$ if $\left(\frac{\kappa-4}{p}\right)=1$.
\end{theorem}
For each such $\kappa$ and for infinitely many primes $p$ (depending on $\kappa$) of natural density at least $1/2$,
there exists an explicit $K_{3,3}$-subdivision in $\Gkp$.

\item[$(2)$]
Moreover, the following theorem implies that for almost all $\kappa$,
the natural density of primes $p$ for which $(1)$ holds is at least $13/16$.
\begin{theorem}
\label{thm:np density}
\begin{itemize}
\item[$(1)$]
For each $\kappa\in\mathbb{Z}\setminus\{4\}$ such that none of
$\eta_{1,\kappa}$,
$\eta_{2,\kappa}$,
$5$, and $\kappa-4$
is a square in $\mathbb{Z}$,
and that the ratio of any two of them is not a square in $\mathbb{Q}$,
we have
\begin{equation}
\label{for:lower bound of Dnp}
\delta_{\kappa}\ge \frac{13}{16}.
\end{equation}
That is, $\Gkp$ contains a $K_{3,3}$-subdivision
for at least $81.25\%$
of primes $p$ asymptotically.
\item[$(2)$]
The natural density
of $\kappa\in\mathbb{Z}\setminus\{4\}$ for which $(1)$ holds
is equal to $1$.
\end{itemize}
\end{theorem}

\end{enumerate},

Question: Let \(\kappa\in\mathbb Z\setminus\{4\}\). For a prime \(p>3\), let \(\mathcal M_\kappa(p)\subset \mathbb F_p^3\) be the set of solutions to \(x^2+y^2+z^2=xyz+\kappa\), and let \(\mathcal G_\kappa(p)\) be the graph with vertex set \(\mathcal M_\kappa(p)\), where two vertices are adjacent exactly when one is obtained from the other by a Vieta involution \(R_1(x,y,z)=(yz-x,y,z)\), \(R_2(x,y,z)=(x,zx-y,z)\), or \(R_3(x,y,z)=(x,y,xy-z)\). Write \(\left(\frac{\cdot}{p}\right)\) for the Legendre symbol, and let \(\xi_{1,\kappa},\eta_{1,\kappa},\xi_{2,\kappa},\eta_{2,\kappa}\) denote the auxiliary integers attached to \(\kappa\). Which statement about subdivisions of \(K_{3,3}\) in \(\mathcal G_\kappa(p)\) holds?

An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) and \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(13/16\) (equivalently, at least \(81.25\%\) of primes asymptotically), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).

Choices:
B. An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) and \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=-1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(13/16\), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).
C. If at least one of the following holds for a prime \(p>3\): (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) together with \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=1\); or (iii) \(\left(\frac{\kappa-4}{p}\right)=1\), then \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision. Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), there are infinitely many primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision.
D. An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) with \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(13/16\), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).
E. An explicit \(K_{3,3}\)-subdivision exists in \(\mathcal G_\kappa(p)\) whenever at least one of the following holds: (i) \(\xi_{1,\kappa}\eta_{1,\kappa}\neq 0\) and \(\left(\frac{\eta_{1,\kappa}}{p}\right)=1\); (ii) \(\xi_{2,\kappa}\eta_{2,\kappa}\neq 0\), \(5(\kappa^2-73\kappa+61)\neq 0\), and either (A) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=1\) and \(\left(\frac{5}{p}\right)=\left(\frac{8\kappa-17+2\sqrt{\eta_{2,\kappa}}}{p}\right)=\left(\frac{8\kappa-17-2\sqrt{\eta_{2,\kappa}}}{p}\right)=1\), or (B) \(\left(\frac{\eta_{2,\kappa}}{p}\right)=-1\) and \(\left(\frac{5}{p}\right)=1\); (iii) \(\left(\frac{\kappa-4}{p}\right)=1\). Consequently, for every \(\kappa\in\mathbb Z\setminus\{4\}\), the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains an explicit \(K_{3,3}\)-subdivision is infinite and has natural density at least \(1/2\). Moreover, if none of \(\eta_{1,\kappa},\eta_{2,\kappa},5,\kappa-4\) is a square in \(\mathbb Z\) and the ratio of any two of them is not a square in \(\mathbb Q\), then the set of primes \(p\) for which \(\mathcal G_\kappa(p)\) contains a \(K_{3,3}\)-subdivision has natural density at least \(3/4\), and the set of integers \(\kappa\in\mathbb Z\setminus\{4\}\) satisfying this arithmetic hypothesis has natural density \(1\).

lower-density bound 13/16 replaced by 3/4

The stem does not explicitly or implicitly identify the correct option. It asks for the valid existence statement, and the alternatives are close enough that the answer is not leaked by wording alone.

This is essentially a theorem-recall item: the student is asked to choose which full existence statement from the paper is correct. It is very close to a direct restatement rather than an application or inference task.

Some careful comparison is required, since the distractors differ by subtle sign changes, omitted subcases, or weakened conclusions. However, the task mainly tests precise recall/textual matching, not genuine generative mathematical reasoning.

The distractors are plausible and well-targeted: they reflect realistic failure modes such as dropping a case, flipping a Legendre-symbol sign, or weakening the theorem. They are distinct without being obviously absurd.

A technically well-constructed recall/comparison MCQ with strong distractors, but it scores poorly on tautology avoidance and only modestly on generative reasoning because it mainly asks for recognition of the paper’s theorem statement.


\label{main} Let $G$ be a primitive permutation group with socle $\hbox{\rm PSU}(3,q)$, where $q=p^m\ge 7$ for an odd prime $p$. Suppose a point-stabilizer
  $M$ of $G$ contains $\hbox{\rm PSO}(3,q)$. Then $b(G)=2$ and either (i) $G=\hbox{\rm PSU}(3,q)$ and $M=\hbox{\rm SO}(3,q)$; or (ii) $G=\hbox{\rm PSU}(3,q)\ZZ_{m_1}$ and  $M=\hbox{\rm SO}(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and  $m_1\ne  1)$. Further, the BG-Conjecture is valid for  case (i);
  and for   $\sqrt{q}\geq 17$ or $45$,  if $3\nmid (q+1)$  and $3\bigm| (q+1)$, respectively in case (ii).

\begin{theorem} \label{main} Let $G$ be a primitive permutation group with socle $\PSU(3,q)$, where $q=p^m\ge 7$ for an odd prime $p$. Suppose a point-stabilizer
  $M$ of $G$ contains $\PSO(3,q)$. Then $b(G)=2$ and either (i) $G=\PSU(3,q)$ and $M=\SO(3,q)$; or (ii) $G=\PSU(3,q)\ZZ_{m_1}$ and  $M=\SO(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and  $m_1\ne  1)$. Further, the BG-Conjecture is valid for  case (i);
  and for   $\sqrt{q}\geq 17$ or $45$,  if $3\nmid (q+1)$  and $3\di (q+1)$, respectively in case (ii).
  \end{theorem}

A {\em base} for a finite permutation group $G$ on a set $\Omega$ is a subset $\Delta\subset \Omega$ whose pointwise  stabilizer $G_{(\Delta)}$ is trivial.  This  concept  is a natural generalization of basis for  vector spaces. The {\em base size} $b(G)$ of $G$  is the minimal cardinality $|\Delta|$, where $\Delta$ goes  over all bases for $G$, while  a  base $\Delta '$ with $|\Delta '|=b(G)$  is called a {\em base size set}. With a history dating to the 19th century, these concepts remain vital to contemporary algebra and combinatorics; see \cite{Bailey-Cameron,LS} and \cite[$\S5$]{B} for overviews and \cite{G,S} for applications.

Let $G\leq \hbox{\rm Sym}(\Omega)$ and $\Delta$  a base for $G$. If $x,y\in G$, then $\alpha^x=\alpha^y$ for all $\alpha\in\Delta$ if and only if $x=y$. This implies that each group element is uniquely determined by its action on a base $\Delta$. In particular, if $\Omega$ is finite, then $|G|\leq |\Omega|^{b(G)}$. The problem of bounding $|G|$ in terms of $|\Omega|$, which attracted significant interest in the 19th century, motivated early investigations of bases. In 1992, Blaha \cite{Ba} proved that determining whether or not $G$ has a base size at most a given constant is an NP-complete problem.
 Having established the O'Nan-Scott theorem, which characterizes finite primitive groups through the structure and action of their socle, there has been a surge of research into the base sizes of primitive permutation groups. This surge was catalyzed by the resolution of several highly influential conjectures by Babai, Cameron, Kantor, and Pyber from the 1990s, spurring new research directions as evidenced by \cite{BGS1,BGS2,BLS,BBW,DHM,Fawcett1,Fawcett2,Ha,H,S1}.

Primitive permutation groups with a base of size $1$ are cyclic groups $\ZZ_p$, which makes the family of finite primitive groups with a base of size 2 particularly significant due to their notably small base.
Saxl's ``base-two programme", which aims to classify finite primitive permutation groups with base size $2$, has inspired a great deal of research. To study the bases of permutation groups $G$ with $b(G)=2$, Burness and Giudici \cite{BG} introduced the {\em Saxl graph}, which is defined in terms of these groups. Let $G$ be a permutation group on $\Omega$ with $b(G)=2$. The vertex set of a Saxl graph $\Sigma(G)$ (denoted by $\Sigma$, simply) of $G$ is just
$\Omega$, while two vertices are adjacent if and only if they form a base. Clearly, if $G$ is a transitive permutation group, then the Saxl graph graph $\Sigma(G)$ is vertex-transitive. Moreover, if $G$ is primitive, then $\Sigma(G)$ is connected, but the converse is not true.
  In \cite{BG}, the authors discussed  the valency, connectivity, hamiltonicity and the independence number of $\Sigma(G)$, and   proposed the following conjecture.
\vskip -10pt
\begin{conj}{\rm \cite{BG}} \label{BG}
Let $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\Sigma(G)$ have a common neighbour.
\end{conj}
\vskip -5pt

 From now on, we assume that $G\leq \hbox{\rm Sym}(\Omega)$ acts transitively on $\Omega$. Fixing a point $\alpha\in \Omega$, we call the orbits of $G_\alpha$ the {\em suborbits} of $G$ relative to $\alpha$; the trivial suborbit is $\{\alpha\}$ itself.
   Now $\{\alpha,\beta\}$ is a base for $G$ with $b(G)=2$ if and only if $G_{\alpha}$ acts regularly on the suborbit containing $\beta$.
   It follows that the neighborhood $\si_1(\alpha)$ of $\alpha$ in
  $\Sigma(G)$ is the union $\Gamma$ of all regular suborbits of $G$ relative to $\alpha$. Consequently,
  Conjecture~\ref{BG} can be restated in terms of $\Gamma$ as follows, which we refer to as the BG-Conjecture.
\begin{conj}$($BG-Conjecture$)$ \label{BG1}
 Every finite primitive permutation group $G$ with $b(G)=2$ has the property  that
   $\Gamma\cap \Gamma^g\ne \emptyset$ for any $g\in G$, where $\Gamma$ is the union of all regular suborbits of $G$ relative to a point $\alpha$.
\end{conj}
\vskip -5pt

Using a probabilistic approach, Burness and Giudici \cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \cite{HPHD}).
Furthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \cite{BG} also confirms the conjecture for many sporadic simple groups.
Burness and Huang \cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \cite{BH2}.
Lee and Popiel \cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs
and Saxl hypergraphs in
\cite{FH, LP1}.

So far this conjecture remains open for all  primitive  groups $G$ whose socle is a simple group of  Lie-type of rank $1$. Our research aimed to tackle this conjecture in case of such groups of rank one.
  This class of groups contains four families, with  respective socles $\hbox{\rm PSL}(2,q)$,   $\hbox{\rm PSU}(3,q)$, $\hbox{\rm Ree}(q)$ and $\hbox{\rm Sz}(q)$.
Having established the conjecture for groups with socle $\hbox{\rm PSL}(2,q)$ in \cite{BH,Chen-Du},
  we now address the case of $\hbox{\rm PSU}(3,q)$ in a two-part series. In this paper (Part I), we prove the conjecture for primitive groups $G$ with this socle, whose point stabilizers contain $\hbox{\rm PSO}(3,q)$, with a few small exceptions (see Therorem~\ref{main}). Part II (i.e.~\cite{Chen-Du-Li1}) will treat the remaining primitive actions for $G$. Part III (i.e., \cite{Chen-Du2}) will present the cases  $\hbox{\rm soc}(G)\in \{\hbox{\rm Ree}(q)$, $\hbox{\rm Sz}(q)$\}.
We now state the main theorem of this paper.

Let $G\leq \hbox{\rm Sym}(\Omega)$ and $\Delta$ a base for $G$. If $x,y\in G$, then $\alpha^x=\alpha^y$ for all $\alpha\in\Delta$ if and only if $x=y$. This implies that each group element is uniquely determined by its action on a base $\Delta$. In particular, if $\Omega$ is finite, then $|G|\leq |\Omega|^{b(G)}$. The problem of bounding $|G|$ in terms of $|\Omega|$, which attracted significant interest in the 19th century, motivated early investigations of bases. In 1992, Blaha \cite{Ba} proved that determining whether or not $G$ has a base size at most a given constant is an NP-complete problem.
 Having established the O'Nan-Scott theorem, which characterizes finite primitive groups through the structure and action of their socle, there has been a surge of research into the base sizes of primitive permutation groups. This surge was catalyzed by the resolution of several highly influential conjectures by Babai, Cameron, Kantor, and Pyber from the 1990s, spurring new research directions as evidenced by \cite{BGS1,BGS2,BLS,BBW,DHM,Fawcett1,Fawcett2,Ha,H,S1}.

Primitive permutation groups with a base of size $1$ are cyclic groups $\ZZ_p$, which makes the family of finite primitive groups with a base of size 2 particularly significant due to their notably small base.
Saxl's ``base-two programme", which aims to classify finite primitive permutation groups with base size $2$, has inspired a great deal of research. To study the bases of permutation groups $G$ with $b(G)=2$, Burness and Giudici \cite{BG} introduced the {\em Saxl graph}, which is defined in terms of these groups. Let $G$ be a permutation group on $\Omega$ with $b(G)=2$. The vertex set of a Saxl graph $\Sigma(G)$ (denoted by $\Sigma$, simply) of $G$ is just
$\Omega$, while two vertices are adjacent if and only if they form a base. Clearly, if $G$ is a transitive permutation group, then the Saxl graph graph $\Sigma(G)$ is vertex-transitive. Moreover, if $G$ is primitive, then $\Sigma(G)$ is connected, but the converse is not true.
 In \cite{BG}, the authors discussed the valency, connectivity, hamiltonicity and the independence number of $\Sigma(G)$, and proposed the following conjecture.
\vskip -10pt
\begin{conj}{\rm \cite{BG}} \label{BG}
Let $G$ be a finite primitive permutation group with $b(G)=2$. Then every two vertices in $\Sigma(G)$ have a common neighbour.
\end{conj}
\vskip -5pt

From now on, we assume that $G\leq \hbox{\rm Sym}(\Omega)$ acts transitively on $\Omega$. Fixing a point $\alpha\in \Omega$, we call the orbits of $G_\alpha$ the {\em suborbits} of $G$ relative to $\alpha$; the trivial suborbit is $\{\alpha\}$ itself.
 Now $\{\alpha,\beta\}$ is a base for $G$ with $b(G)=2$ if and only if $G_{\alpha}$ acts regularly on the suborbit containing $\beta$.
 It follows that the neighborhood $\si_1(\alpha)$ of $\alpha$ in
 $\Sigma(G)$ is the union $\Gamma$ of all regular suborbits of $G$ relative to $\alpha$. Consequently,
 Conjecture~\ref{BG} can be restated in terms of $\Gamma$ as follows, which we refer to as the BG-Conjecture.
\begin{conj}$($BG-Conjecture$)$ \label{BG1}
 Every finite primitive permutation group $G$ with $b(G)=2$ has the property that
 $\Gamma\cap \Gamma^g\ne \emptyset$ for any $g\in G$, where $\Gamma$ is the union of all regular suborbits of $G$ relative to a point $\alpha$.
\end{conj}
\vskip -5pt

Using a probabilistic approach, Burness and Giudici \cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \cite{HPHD}).
Furthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \cite{BG} also confirms the conjecture for many sporadic simple groups.
Burness and Huang \cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \cite{BH2}.
Lee and Popiel \cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs
and Saxl hypergraphs in
\cite{FH, LP1}.

So far this conjecture remains open for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$. Our research aimed to tackle this conjecture in case of such groups of rank one.
 This class of groups contains four families, with respective socles $\hbox{\rm PSL}(2,q)$, $\hbox{\rm PSU}(3,q)$, $\hbox{\rm Ree}(q)$ and $\hbox{\rm Sz}(q)$.
Having established the conjecture for groups with socle $\hbox{\rm PSL}(2,q)$ in \cite{BH,Chen-Du},
 we now address the case of $\hbox{\rm PSU}(3,q)$ in a two-part series. In this paper (Part I), we prove the conjecture for primitive groups $G$ with this socle, whose point stabilizers contain $\hbox{\rm PSO}(3,q)$, with a few small exceptions (see Therorem~\ref{main}). Part II (i.e.~\cite{Chen-Du-Li1}) will treat the remaining primitive actions for $G$. Part III (i.e., \cite{Chen-Du2}) will present the cases $\hbox{\rm soc}(G)\in \{\hbox{\rm Ree}(q)$, $\hbox{\rm Sz}(q)$\}.
We now state the main theorem of this paper.

Using a probabilistic approach, Burness and Giudici \cite{BG} verified the conjecture for a range of almost simple groups. For instance, it holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabilizer is primitive. Their results also cover certain primitive groups of diagonal type and twisted wreath products of sufficiently large order, which have been extended recently by Huang (see Sections 5.8 and 5.9 of \cite{HPHD}).
Furthermore, by employing the Magma database, they verified the conjecture for all primitive groups of degree at most $4095$. The paper \cite{BG} also confirms the conjecture for many sporadic simple groups.
Burness and Huang \cite{BH} later proved the conjecture for almost simple primitive groups with soluble point stabilisers; additionally, they studied it for primitive groups of product type in a separate paper \cite{BH2}.
Lee and Popiel \cite{LP} have recently proved the conjecture for most affine-type groups with sporadic point stabilizers. The theory was subsequently extended with the introduction of generalized Saxl graphs
and Saxl hypergraphs in
\cite{FH, LP1}.

Remark that for the reason of the length of the paper, we are not able to improve the lower bound $\sqrt{q}\geq 17$ (resp. $45$) for small exceptions in case (ii) of Theorem~\ref{main}.
 Hopefully, these small cases may be verified directly with a sufficiently powerful computer.

Combining Theorem~\ref{main} and the main results of \cite{BH,Chen-Du,Chen-Du2}, we get
\begin{theorem} \label{main'} Let $G$ be a primitive group whose socle is a simple group of Lie-type of rank $1$ such that $b(G)=2$.
 Then the BG-Conjecture is valid, except for $G=\PSU(3,q)\ZZ_{m_1}$ and $M=\SO(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and $m_1\ne 1)$, where
 $\sqrt{q}\le 17$ or $45$, if $3\nmid (q+1)$ and $3\di (q+1)$, respectively.
 \end{theorem}

\section{$\PSL(3,q)\lvertneqq G$ and $\SO(3,q)\lvertneqq M$}\label{ell}
 In this section, we mainly prove the following theorem.
\vskip -7pt \begin{theorem} \label{main2} Let $G=\PSU(3,q)\ZZ_{m_1}$ and
 $\a=M=\PSO(3,q)\ZZ_{m_1}\le G$, with $q=p^m\ge 7$ for an odd prime $p$, $m_1\mid 2m$ and $m_1\ne 1$.
Under the primitive right multiplication action of $G$ on $\Omega=[G:M]$, we have $b(G)=2$ and the BG-Conjecture is true, provided $\sqrt{q}\geq 17$ or $45$ if $3\nmid (q+1)$ and $3\di (q+1)$, respectively.
 \end{theorem}

\section{A lower bound of $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|$}
 Theorem~\ref{main21} is the main result of this section.
\begin{theorem}\label{main21}
Using the notation from Section~\ref{ell},
consider the right multiplication action of $T$ on the cosets $[T:M_0]$, where $d=(3, q+1)$ and $q\geq 7$ is odd. Suppose that $\sqrt{q}\geq 15d$. Then
$|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|\ge \frac{4}{9}q^5-\frac{1}{3}q^4\sqrt{q}-22.1555q^4$ for $d=1$; and
$|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|\ge \frac{4}{27}q^5-\frac{31}{27}q^4\sqrt{q}-16.8372q^4$ for $d=3$.
\end{theorem}
\demo From now on, set
$G=T=\PSU(3,q)$ and $M=M_0=\PSO(3,q)$. Since $M$ is maximal in $G$, we identify each coset $Mg$ with its point stabilizer $M^g$ so that $\O=\{ M^g\mid g\in G\}$, with the conjugacy action. Set $\a=M$.
 Let $\a':=M'=M^{g'}$ for any $g'\in G$. Then we shall get a lower bound of $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|$.
 \vskip 3mm
The possible isomorphic types for $M\cap M'$ are:
 \vspace{-5pt}\begin{equation}\label{A_i} A_q=\ZZ_p^m.\ZZ_2,\, A_{q\pm 1}=\D_{2(q\pm 1)}, \, A_3=\D_4, A_4=\Alt(4)\, ({\rm if}\, 3\mid q+1), A_2=\ZZ_2 \, {\rm or}\, A_1=1.\end{equation}
All of these groups $A_\imath$ contain at least one involution, except for $A_1$. Therefore, set
\vspace{-5pt}\begin{equation}\label{I} I=\{ g\in M\mid |g|=2\}\quad {\rm and}\quad I'=\{ g\in M'\mid |g|=2\}.\end{equation}
Clearly, $0 \le |I \cap I'| \le q+2$, giving
 \vspace{-5pt}\begin{equation}\label{lw}\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')=\{M_\imath\in \O \setminus \{M, M'\}\mid t, t'\in M_\imath,\,\, {\rm for \, some}\,\, t\in I, t'\in I' \}.\end{equation}
 For the reasen of a simplicity, we turn to consider a subset ${\cal W}$ of $\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')$, where
 \vspace{-5pt}\begin{equation}\label{W_1} \cal W=\{ M_\imath\in \O \setminus \{M, M'\}\mid t, t'\in M_\imath\,\, {\rm for \, some}\, \, t\in I, t'\in I' \,\, {\rm but}\,\, t, t'\notin I\cap I'\}.\end{equation}
To determine $|{\cal W}|$, we introduce a set of incident triples (under the inclusion relation)
 \vspace{-5pt}\begin{equation}\label{E} \cal E=\{(t, M_\imath, t')\mid M_\imath\in {\cal W}, t\in M\cap M_\imath, t'\in M'\cap M_\imath, t, t'\notin I\cap I'\},\end{equation}
\f while for any given $ t\in I\setminus I'$ and $M_\imath\in {\cal W}$, we introduce two respective sets of pairs
\vspace{-5pt}\begin{equation}\label{E(t)} \cal E(t)=\{(M_\imath,t')\mid (t, M_\imath, t')\in \cal E\}\,\, {\rm and}\,\, \cal E(M_\imath)=\{(t,t')\mid (t, M_\imath, t')\in \cal E\},\end{equation}
so that we may count $|\cal E|$ by two different ways, that is,
\vspace{-5pt}\begin{equation} \label{2way} \sum_{t\in I\setminus I'}|\cal E(t)|=|\cal E|=\sum_{M_\imath\in {\cal W}}|\cal E(M_\imath)|.\end{equation}
Further, according to $M_\imath\cap M\cong A_j$, $M_\imath\cap M'\cong A_k$, $M_\imath\cap M\cap M'\cong A_l$, where $j, k,l\in \{1,2,3,4,q,q\pm1\}$, as mentioned in Eq(\ref{A_i}),
 the set ${\cal W}$ is the union of the following subsets ${\cal W}_{j, k, l}$, where
\vspace{-5pt}\begin{equation}\label{jkl}\begin{array}{lll}
 {\cal W}_{j,k, l}&=&\{M_\imath\in {\cal W}\mid M_\imath\cap M\cong A_j, M_\imath\cap M'\cong A_k, M_\imath\cap M\cap M'=A_l\}\end{array}\end{equation}
and $j,k,l\in\{1,2,3,4,q,q\pm1\}$,
reminding that $(M_\imath\cap M) \cap (M_\imath\cap M')=M\cap M_\imath\cap M'$. Moreover, set
\vspace{-5pt}\begin{equation}\label{n}
 n_{j, k, l}=|{\cal W}_{j,k,l}|.\end{equation}
 Then by Eq(\ref{2way}) and Eq(\ref{jkl}), we have
 \vspace{-5pt}$$\begin{array}{ll}
 &\sum_{t\in I\setminus I'}|\cal E(t)|=\sum_{j,k, l} n_{j,k, l} |\{\cal E(M_\imath): M_\imath\in {\cal W}_{j,k, l}\}|,
 \end{array}$$
\begin{equation}\label{w1}
\begin{array}{lll}|{\cal W}|&=&\sum\limits_{j,k, l}n_{j,k, l}=\sum\limits_{t\in I\setminus I'}|\cal E(t)|
-(\sum\limits_{t\in I\setminus I'}|\cal E(t)|-\sum\limits_{j,k, l}n_{j,k,l})\\
&=&\sum\limits_{t\in I\setminus I'}|\cal E(t)|
-\sum\limits_{j,k, l} n_{j,k,l}(|\{\cal E(M_\imath): M_\imath\in {\cal W}_{j,k, l}\}|-1)\\
&=&{\hm A}-{\hm B}-{\hm C}-{\hm D},\quad {\rm where} \end{array}
\end{equation}
$$\begin{array}{ll}
&{\hm A}=\sum\limits_{t\in I\setminus I'}|\cal E(t)|,\hskip 2cm
{\hm B}=\sum\limits_{4, q\not\in \{j, k\}}n_{j, k, 1}(|\{\cal E(M_\imath): M_\imath\in{\cal W}_{j,k,1}\}|-1),\\
&{\hm C}=\sum\limits_{4, q\not\in \{j, k\},l\ne 1}n_{j, k, l}(|\{\cal E(M_\imath): M_\imath\in{\cal W}_{j,k,l}\}|-1),\\
&{\hm D}=\sum\limits_{\{4, q\}\cap \{j, k\}\ne \emptyset} n_{j, k, l}(|\{\cal E(M_\imath): M_\imath\in{\cal W}_{j,k, l}\}|-1).\\
\end{array}$$
\f In the following Lemmas~\ref{AB},~\ref{C} and~\ref{D} and~\ref{DD}, it will be respectively shown that
\vspace{-5pt}$$\begin{array}{lll}
{\hm A}&\geq &q^5-1.0355 q^4,\, \hbox{$d=1;$}\,\, \frac 13q^5-0.3342 q^4, \, \hbox{$d=3;$}\,\\
{\hm B} &\leq & \frac{5}{9}q^5+\frac{1}{3}q^4\sqrt{q}+15.25 q^4, \, \hbox{$d=1$;} \,
 \frac{5}{27}q^5+\frac{31}{27}q^4\sqrt{q}+10.74 q^4, \, \hbox{$d=3$;}\\
 {\hm C}&\leq& 3.85q^4, \hbox{$d=1$;} \, 3.26 q^4, \hbox{$d=3$};\\
 {\hm D}&\leq& 2.02 q^4, \, \hbox{$d=1$;} \, 2.503 q^4,\, \hbox{$d=3$}. \\
\end{array}$$
\f By using Eq(\ref{lw}), Eq(\ref{W_1}) and Eq(\ref{w1}), we get $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|\ge|{\cal W}|= {\hm A}-{\hm B}-{\hm C}-{\hm D}$. Inserting these bounds of ${\hm A}$, ${\hm B}$, ${\hm C}$, ${\hm D}$, we get a lower bound for $|\Gamma_{nr}(\a)\cap \Gamma_{nr}(\a')|$ as shown in Theorem~\ref{main21}, omitting the details.
\qed

Remark that for the reason of the length of the paper, we are not able to  improve the lower bound  $\sqrt{q}\geq 17$ (resp. $45$) for small exceptions in case (ii) of Theorem~\ref{main}.
 Hopefully,  these  small cases may be verified directly with a sufficiently powerful computer.

Combining Theorem~\ref{main} and the main results of \cite{BH,Chen-Du,Chen-Du2}, we get
\begin{theorem} \label{main'} Let $G$ be  a primitive  group whose socle is a simple group of  Lie-type of rank $1$ such that $b(G)=2$.
 Then  the BG-Conjecture is valid, except for $G=\hbox{\rm PSU}(3,q)\ZZ_{m_1}$ and  $M=\hbox{\rm SO}(3,q)\ZZ_{m_1}$ $(m_1\mid 2m$ and  $m_1\ne  1)$, where
    $\sqrt{q}\le 17$ or $45$,  if $3\nmid (q+1)$  and $3\bigm| (q+1)$, respectively.
  \end{theorem}

\vskip 3mm
\noindent {\bf  Outline  of the proof of Theorem~\ref{main}}: From now on, let $G$ be an almost simple group with the socle $T=\hbox{\rm PSU}(3,q)$, so that $T\leq G\leq \hbox{\rm Aut\,}(T)$, where $q=p^m$.
 Let $\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\hbox{\rm PSO}(3,q)$, up to conjugacy. Consider the primitive permutation representations of $G$ on the right cosets $[G:M]$.
 Then $q\ge7$ and either: (i) $G=T$ and $M=M_0$; or (ii) $G=T\ZZ_{m_1}$ and  $M=M_0\ZZ_{m_1}$, where $m_1\mid 2m$ and $m_1\ne 1$.
For the first case, we exhibit a common  neighbor of $\alpha $ and $\alpha^g$ in the Saxl graph for any $g\in G$ via a direct geometric construction in Section 2.
However, we are not able to extend  this geometric construction   to the second case, which necessitates a new approach developed in Section 3.
Consequently, the proof of Theorem~\ref{main} is obtained by combining Theorems~\ref{main1} and \ref{main2}.
Moreover, to prove Theorem~\ref{main2} crucially depends on estimates of $|\Gamma_r(\alpha)|$ and $|\Gamma_{nr}(\alpha) \cap \Gamma_{nr}(\alpha')|$, developed in Sections 4 and 5 respectively. Recall that $\Gamma_r(\alpha)$ (resp. $\Gamma_{nr}(\alpha)$) is the union of all regular (resp. non-regular) suborbits of $G$ relative to $\alpha$.
\vskip -7pt

Outline of the proof of Theorem~\ref{main}: Let $G$ be almost simple with socle $T=\hbox{\rm PSU}(3,q)$, so $T\le G\le \hbox{\rm Aut}(T)$, $q=p^m$. Let $\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\hbox{\rm PSO}(3,q)$ (up to conjugacy), and consider the primitive action of $G$ on the right cosets $[G\!:\!M]$. Then $q\ge 7$ and either (i) $G=T$ and $M=M_0$, or (ii) $G=T\ZZ_{m_1}$ and $M=M_0\ZZ_{m_1}$ with $m_1\mid 2m$ and $m_1\ne 1$. In case (i), a common neighbor of $\alpha$ and $\alpha^g$ in the Saxl graph (for any $g\in G$) is exhibited by a direct geometric construction in Section 2. This geometric construction does not extend to case (ii), so a new approach is developed in Section 3. Theorem~\ref{main} then follows by combining Theorems~\ref{main1} and~\ref{main2}. The proof of Theorem~\ref{main2} depends crucially on estimates for $|\Gamma_r(\alpha)|$ and $|\Gamma_{nr}(\alpha)\cap \Gamma_{nr}(\alpha')|$, developed in Sections 4 and 5, where $\Gamma_r(\alpha)$ (resp. $\Gamma_{nr}(\alpha)$) is the union of regular (resp. non-regular) suborbits of $G$ relative to $\alpha$.

Outline of the proof of the main theorem: Let $G$ be almost simple with socle $T=\hbox{\rm PSU}(3,q)$, so $T\le G\le \hbox{\rm Aut}(T)$, $q=p^m$. Let $\alpha=M$ be a maximal subgroup of $G$ containing $M_0=\hbox{\rm PSO}(3,q)$ (up to conjugacy), and consider the primitive action of $G$ on the right cosets $[G\!:\!M]$. Then $q\ge 7$ and either (i) $G=T$ and $M=M_0$, or (ii) $G=T\ZZ_{m_1}$ and $M=M_0\ZZ_{m_1}$ with $m_1\mid 2m$ and $m_1\ne 1$. In case (i), a common neighbor of $\alpha$ and $\alpha^g$ in the Saxl graph (for any $g\in G$) is exhibited by a direct geometric construction developed earlier. This geometric construction does not extend to case (ii), so a new approach is developed later.

We first prove the following theorem.
\begin{theorem} \label{main1}  Let $G=\PSU(3,q)$ and $M=\PSO(3,q),$ a maximal subgroup of $G$, where $q=p^m\ge 7$ for an odd prime $p$.
Consider the primitive right multiplication action  of $G$ on the set of right cosets $\Omega =[G:M]$.  Then $b(G)=2$ and the BG-Conjecture holds.
  \end{theorem}

We also prove the following theorem.
\begin{theorem} \label{main2}  Let $G=\PSU(3,q)\ZZ_{m_1}$ and
    $\a=M=\PSO(3,q)\ZZ_{m_1}\le G$, with  $q=p^m\ge 7$ for an odd prime $p$,  $m_1\mid 2m$ and $m_1\ne 1$.
Under the primitive right multiplication action of $G$ on $\Omega=[G:M]$, we have $b(G)=2$ and the BG-Conjecture is true, provided $\sqrt{q}\geq 17$ or $45$ if $3\nmid (q+1)$ and $3\di (q+1)$, respectively.
  \end{theorem}

In establishing the main theorem, we combine Theorem~\ref{main1} and Theorem~\ref{main2}. The proof of Theorem~\ref{main2} depends crucially on estimates for $|\Gamma_r(\alpha)|$ and $|\Gamma_{nr}(\alpha)\cap \Gamma_{nr}(\alpha')|$, developed later, where $\Gamma_r(\alpha)$ (resp. $\Gamma_{nr}(\alpha)$) is the union of regular (resp. non-regular) suborbits of $G$ relative to $\alpha$.

Question: Let $G\leq \mathrm{Sym}(\Omega)$ be a finite primitive permutation group with socle $\mathrm{PSU}(3,q)$, where $q=p^m\ge 7$ for an odd prime $p$, and let $M=G_\alpha$ be a point-stabilizer containing $\mathrm{PSO}(3,q)$. Here $b(G)$ is the minimal size of a base, i.e. a subset $\Delta\subseteq \Omega$ whose pointwise stabilizer in $G$ is trivial. When $b(G)=2$, the Saxl graph $\Sigma(G)$ is the graph with vertex set $\Omega$ in which two vertices are adjacent exactly when they form a base; the BG-Conjecture asserts that every two vertices of $\Sigma(G)$ have a common neighbour. Under these assumptions, which statement about $G$, $M$, $b(G)$, and the BG-Conjecture holds?

One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$ with $m_1\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $\sqrt q\ge 17$ if $3\nmid(q+1)$, and whenever $\sqrt q\ge 45$ if $3\mid(q+1)$.

Choices:
B. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$ with $m_1\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $q\ge 17$ if $3\nmid(q+1)$, and whenever $q\ge 45$ if $3\mid(q+1)$.
C. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$ with $m_1\ne 1$.
D. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid 2m$. Moreover, the BG-Conjecture holds in both cases (i) and (ii).
E. One has $b(G)=2$, and either (i) $G=\mathrm{PSU}(3,q)$ and $M=\mathrm{SO}(3,q)$, or (ii) $G=\mathrm{PSU}(3,q)\mathbb Z_{m_1}$ and $M=\mathrm{SO}(3,q)\mathbb Z_{m_1}$ for some $m_1\mid m$ with $m_1\ne 1$. Moreover, the BG-Conjecture holds in case (i); and in case (ii) it holds whenever $\sqrt q\ge 17$ if $3\nmid(q+1)$, and whenever $\sqrt q\ge 45$ if $3\mid(q+1)$.

The stem does not explicitly reveal the correct option. It states the setup and asks for the full classification, but the exact divisibility condition and BG-Conjecture thresholds must still be selected from the choices.

This is essentially a direct recall of a specific classification theorem and its corollary. The question asks for the theorem statement itself rather than requiring application to a new situation.

There is some pressure to distinguish subtle variants in the hypotheses and conclusions, such as whether m1 divides 2m or m, whether m1 ≠ 1 is required, and the conditional BG thresholds. However, this is mainly theorem recognition/recall rather than genuine generative mathematical reasoning.

The distractors are plausible and carefully constructed: they alter boundary conditions, omit a necessary exclusion, weaken the BG conclusion, or overstate it. These align with realistic mathematical errors and are not trivially dismissible.

A technically well-constructed recall question with strong distractors, but it is largely theorem restatement rather than a true reasoning-based MCQ.