\label{th-main}
For each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.
Moreover, { if $\Ga$ is a $k$-homogeneous graph with $k\geqslant 2$, then} one of the following holds:
\begin{itemize}
\item [{\rm(1)}] $k\geqslant 5$, and $\Ga$ is one of { $\C_5 (\cong\overline{\C_5})$, $\K_3\square\K_3 (\cong \overline{\K_3\square\K_3})$, $n\K_m$, and $\K_{n[m]}$; furthermore, $\Ga$ is homogeneous.}

{ \item[(2)] $\Ga$ is either the Schl\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\Ga$ is $4$-homogeneous but not $5$-homogeneous.

\item[(3)] $\Ga$ is either $\K_n\square\K_n$ with $n\geqslant4$, or a graph from Theorem~\ref{th:3sethomo}(1)--(4), or the complement of any such graph; furthermore, $\Ga$ is $3$-homogeneous but not $4$-homogeneous.

\item[(4)] $\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\Ga$ is $2$-homogeneous but not $3$-homogeneous. }

\end{itemize}

\begin{array}{cllllc}
\hline
\text{Line}&S& S_\alpha & \text{degree} &\text{subdegrees}& \text{rank of }\N_{{\rm Sym}(\Om)}(S) \\
\hline
48 &  \mathrm{P\Omega}_7(3)  &  \PSp_6(2)  & 3159 &  1,288,630,2240  &  4 \\
49 &  \mathrm{P\Omega}^{+}_8(2)  &  \A_9  & 960 &  1,84,315,560  &  4 \\
50 & \mathrm{P\Omega}^{+}_8(3)  &  \mathrm{P\Omega}^{+}_8(2)  &  28431  &  1,960,960,960,3150,22400  &  4 \\
\hline
\end{array}
\]
It implies that all orbitals of $G$ are self-paired.
\medskip

\f{\bf Case~5.}\ The Exceptional Group of Lie Type.\medskip

In this case, $(S, S_\alpha)$ is one of the pairs in Lines 51--68 of \cite[Table~I]{Cuypers}.

 \vspace{0.1cm}
\underline{Lines  51--54, 58--59, 61, 63, 65, 67--68}:\  These lines are impossible by Lemma~\ref{lm:parabolic}.

 \vspace{0.1cm}
\underline{Lines 55--56}:\
Now $S=G_2(q)$ and $S_\alpha=\mathrm{SU}_3(q).2$ or $\SL_3(q).2$.
By \cite{Liebeck-2-closures}, all orbitals of $S$ are self-paired.

 \vspace{0.1cm}
\underline{Line  57}:\
Now $S=G_2(4)$ and $S_\alpha=\mathrm{J}_2$, which  gives  rise to a group of rank $3$ by~\cite{Liebeck-Saxl}.

 \vspace{0.1cm}
\underline{Line  60}:\
Now $S={}^2F_4(2)'$ and $S_\alpha=\PSL_3(2).2$. By computation in Magma~\cite{BCP},  $S$ has rank $4$ with subdegrees $1$, $312$, $351$ and $936$.

 \vspace{0.1cm}
\underline{Line  62}:\
Now $S=F_4(2)$ and $S_\alpha=\PSp_8(2)$. From Atlas~\cite{Conway-Atlas} we see that $G=S$.
 By~\cite[Lemma~8.6~(i)]{Cuypers}, $G$ would have  rank $5$.

 \vspace{0.1cm}
\underline{Line 64}:\
Now $S={}^2E_6(2)$ and $S_\alpha=F_4(2)$. By \cite[Lemma~8.6(ii)]{Cuypers}, $G$ has rank $4$. In this case, $S_\alpha$ is the centraliser of the out-involution $\s$ of $S$. Let $G={}^2E_6(2){:}\lg\s\rg$ and $C=F_4(2)\times\lg\s\rg$. Set $\Om=\{\s^g \mid g\in G\}$. Now let $H\leq G$ s.t. $H=\PSU_6(2){:}\lg\s\rg$. Then $C_H(\s)=\PSp_6(2)\times\lg\s\rg$. Take $x,y\in\Om$ such that $x,y$ are in the same orbit of $C$. Then $\s x$ and $\s y$ are conjugate, and so they have the same order. Let $X=\{\s^h\ |\ h\in H\}$. Then $H$ is primitive on $X$ with rank $4$ and all orbitals are self-paired. By Magma~\cite{BCP}, we can obtain that $|\{o (\s x)\ |\ x\in X\}|=4$, and so each orbital of $H$ on $X$ is contained in exactly one orbital of $G$ on $\Om$. This implies that all orbitals of $G$ are also self-paired.
(We are grateful to Prof. Frank L${\rm \ddot{u}}$beck who kindly helps us to obtain the subdegrees (namely, $1,69615, 6336512, 16707600$) of this group by using GAP. )

 \vspace{0.1cm}
\underline{Line  66}:\
Now $S=E_6(2)$ and $S_\alpha=F_4(2)$, and $G$ contains a graph automorphism. By \cite[Lemma~8.7]{Cuypers}, $G$ would have rank $5$.
\hfill\qed

Now combining Propositions~\ref{prop-sporadic}, \ref{prop-alt} and \ref{prop-sim-lie}, the proof of Theorem~\ref{rank-4} is complete.

\section{Proof of Theorems~\ref{th:2sethomo} and \ref{trans-rank-4}}\label{sec6:proof-main-theorems}

\f{\bf Proof of Theorem~\ref{th:2sethomo}.}\ Let $\Ga=(V,E)$ be a $(G,2)$-set-homogeneous graph.
Suppose that $\Ga$ is neither complete nor empty.
If $G$ is imprimitive on the vertex set $V$, then by Lemma~\ref{imp},
$\Ga\cong m\K_b$ or $\K_{m[b]}$ for some positive integers $m$ and $ b$. Then Theorem~\ref{th:2sethomo}~(2) holds.

Assume that $G$ is primitive on $V$. By Lemma~\ref{rank<=5}, $G$ is of rank at most 5.

If $G$ is of rank $3$, then $\Ga$ is $(G,2)$-homogeneous and all candidates for $G$ are listed in \cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}. Then Theorem~\ref{th:2sethomo}~(3)  holds.

If $G$ is of rank $4$, then exactly one of the orbitals is self-paired. By Corollary~\ref{reduction}, $G$ is affine or almost simple. By Theorem~\ref{affine-rank-4}, $G$ is not affine. If $G$ is almost simple, then by Theorem~\ref{rank-4}, $G\cong\PSU_3(3)$ with stabiliser $\PSL_3(2)$, and the subdegrees are $1,7,7$ and $21$. Then $\Ga$ is either the orbital graph of $G$ of order $36$ and of valency $21$, or the complement of this graph. Furthermore, by Magma~\cite{BCP}, $\Aut(\Ga)={\rm P\Gamma U}_3(3)$ is $2$-homogeneous on $\Ga$. This proves part (4) of our theorem.

If $G$ is of rank 5, then none of the non-trivial orbitals of $G$ is self-paired. From Proposition~\ref{AGL_1(p^r)} we obtain part (5) of our theorem. \hfill\qed

\medskip
\f{\bf Proof of Theorem~\ref{trans-rank-4}.}\ Combining Proposition~\ref{th-rank-4}, Lemma~\ref{non-HA-AS}, Theorem~\ref{affine-rank-4} and Theorem~\ref{rank-4}, we obtain the proof of this theorem.\hfill\qed

\section{Finite $3$-set-homogeneous graphs}\label{set:3sethom}

In this section, we shall classify finite $3$-set-homogeneous graphs, and then prove Theorem~\ref{th:3sethomo}.

\subsection{Preliminaries}\label{subsec:3reduce}
\subsubsection{A reduction}
A finite graph $\Ga$ is \emph{strongly regular} with parameters $(v, k, \lambda, \mu)$ if it has $v$ vertices and is regular with valency $k$ where $0<k<v-1$, and if any two adjacent vertices have $\ld=\ld(\Ga)$ common neighbours and any two non-adjacent vertices have $\mu=\mu(\Ga)$ common neighbours. Note that a strongly regular graph is neither complete nor edgeless.

\begin{lem}\label{lem:prim-3ch-is-srg}
Let $\Ga $ be a finite $(G, 3)$-set-homogeneous graph. Then one of the following holds.
\begin{enumerate}
\item[{\rm(1)}] $\Ga \cong\K_n$ or $n\K_1$;

\item[{\rm(2)}]  $\Ga \cong \K_{m[b]}$ or $m\K_b$ where $m,b\geq 2$;

\item [{\rm (3)}]\ $G$ is primitive on $V(\Ga)$ of rank $3$, and $\Ga$ is strongly regular of diameter $2$. Furthermore, if $\Ga$ is not the pentagon, then either $\Ga$ or its complement $\overline{\Ga}$ has girth $3$.
\end{enumerate}
\end{lem}

\proof By \cite[Theorem~1.1 \& Corollary~1.2]{Zhou-EJC2021}, $\Ga$ is $(G,2)$-homogeneous, and then by Theorem~\ref{th:2sethomo}, we conclude that either parts (1) or (2) hold, or $G$ is primitive on $V(\Ga)$ of rank $3$.

Assume that neither part (1) nor part (2) happens. Then $\Ga$ has diameter $2$ and $G$ is arc-transitive on both $\Ga$ and the complement $\overline{\Ga}$ of $\Ga$. This implies that $\Ga$ is strongly regular. The first assertion of the part (3) is proved. If $\Ga$ has valency $2$, then since $\Ga$ has diameter $2$, it follows that $\Ga$ is the pentagon. If $\Ga$ has valency at least $3$, then either $\Ga$ or its complement $\overline{\Ga}$ has girth $3$. \hfill\qed

By Lemma~\ref{lem:prim-3ch-is-srg}, to prove Theorem~\ref{th:3sethomo}, it suffices to classify those $(G,3)$-set-homogeneous graphs $\Ga$ such that $G$ is primitive on $V(\Ga)$ of rank $3$. The following result can be obtained by the fundamental O'Nan-Scott theorem (see \cite[\textsection{1}]{Liebeck-ONan-Scott}).

\begin{theorem}\label{th:rank3groups}
Let $G$ be a primitive group of rank  $3$.
Then one of the following holds:
\begin{itemize}
\item[\rm (1)]  $T\times T\lhd G\leqslant H\wr\S_2$, where $H$ is a $2$-transitive permutation group of degree $n$ and $\soc(H)=T$ is a nonabelian simple group, and $G$ has production action;
\item[\rm (2)]  $G$ is an affine group;
\item[\rm (3)]  $G$ is an almost simple group.
\end{itemize}
\end{theorem}

\begin{lemma}\label{lm:CaseI}
Suppose that $\Ga$ is a finite $(G,3)$-set-homogeneous graph such that $G$ satisfies Theorem~{\rm\ref{th:rank3groups}~(1)}.
Then $\Ga$ is isomorphic to $\K_n\square\K_n$ or its complement $\K_n\times\K_n$, and $\Ga$ is $3$-homogeneous.
\end{lemma}

\proof Without loss of generality, assume that $H$ is a $2$-transitive permutation group on the set $\Del =\{1,2,\dots, n\}$. Then $V(\Ga)=\Del \times\Del $. Take $\a=(1, 1)\in V(\Ga)$. Then the three orbits of $G$ are
\[\Ome_0=\{\a\}, \Ome_1=\{(1, i), (i,1)\mid i\in\Delta\setminus\{1\}\}, \Ome_2=\{(i,j)\mid i, j\in\Del \setminus\{1\}\}.\]
So $\Ga(\a)=\Ome_1$ or $\Ome_2$. It is easy to see that $\Ga\cong\K_n\square\K_n$ or $\K_n\times\K_n$. By \cite[Corollary~1.2]{Cameron-3-hom}, $\Ga$ is $3$-homogeneous.\hfill\qed

By Lemmas~\ref{lem:prim-3ch-is-srg} and \ref{lm:CaseI}, from now on we will adopt the following hypothesis.

\begin{hypothesis}\label{hyp}
Let $\Ga=(V, E)$ be a finite $(G,3)$-set-homogeneous graph satisfying the following:
\begin{itemize}
  \item $G\leq\Aut(\Ga)$ is primitive on $V$ with rank $3$,
  \item $\soc(G)$ is either abelian or nonabelian simple,
  \item $\Ga$ has girth $3$ and diameter $2$.
\end{itemize}
\end{hypothesis}

We will deal with the case where $\soc(G)$ is abelian in Subsection~\ref{subsec:affine}, and the case where $\soc(G)$ is nonabelian simple in Subsection~\ref{subsec:simple}.

\subsubsection{Some useful lemmas}

The first two lemmas concern the arc-stabilisers of $(G,3)$-set-homogeneous graphs. Let $s$ be a positive integer and let $\Sig$ be a graph. For a subgroup $H$ of $\Aut(\Sig)$, we say that $\Sig$ is {\em $(H, s)$-connected-set-homogeneous} if any two isomorphic connected induced subgraphs of $\Sigma$ on at most $s$ vertices are equivalent under $H$. Clearly, every $(H,s)$-set-homogeneous graph is $(H,s)$-connected-set-homogeneous.
For any vertex $v$ of a graph $\Sig$ of diameter at least two, let $\Sig(v)$ be the vertices of $\Sig$ adjacent to $v$, and let $\Sig_2(v)$ be the vertices of $\Sig$ at distance $2$ from $v$.

\begin{lemma}{\rm \cite[Theorem~1.3~(3)-(4)]{Zhou-EJC2021}}\label{rank}
Let $\Sig$ be a finite $(H, 3)$-connected-set-homogeneous non-complete graph of girth $3$ with $H\leq\Aut(\Sig)$. Then for every edge $\{\a, \b\}$ of $\Sig$, we have the following:
\begin{enumerate}
  \item [{\rm (1)}]\ $H_{\a\b}$ has $s$ orbits on $\Sig(\a)\cap \Sig(\b)$ with equal size, where $s=1, 2, 3$ or $6$.
  \item [{\rm (2)}]\ $H_{\a\b}$ has $t$ orbits on $\Sig(\a)-(\Sig(\a)\cap \Sig(\b))\cup\{\b\}$ with equal size, where $t=1$ or $ 2$.
      \end{enumerate}
\end{lemma}

\begin{lemma}\label{lm:Gab10orbits}
Under Hypothesis~{\rm \ref{hyp}}, let $\{\alpha,\beta\}\in E$. Then $G_{\alpha\beta}$ has at most $12$ orbits on $V(\mathit{\Ga})\setminus\{\alpha,\beta\}$.
\end{lemma}

\begin{proof}By Hypothesis~{\rm \ref{hyp}}, $\Ga$ is a $(G,3)$-set-homogeneous graph. Furthermore, $\Ga$ has diameter $2$ and girth $3$. Note that $V(\mathit{\Ga})\setminus\{\alpha,\beta\}$ is partitioned into the following four subsets:
\[\Ga(\a)\cap\Ga(\b), \Ga(\a)\cap\Ga_2(\b), \Ga_2(\a)\cap\Ga(\b), \Ga_2(\a)\cap\Ga_2(\b).\]
Moreover, $G_{\a\b}$ setwise fixes each of the four subsets.

Since every $(G,3)$-set-homogeneous graph is $(G,3)$-connected-set-homogeneous, by Lemma~\ref{rank} $G_{\a\b}$ has at most $6$ orbits on $\Ga(\a)\cap\Ga(\b)$, and has at most $2$ orbits on $\Ga(\a)\cap\Ga_2(\b)$ and on $\Ga_2(\a)\cap\Ga(\b)$.

To complete the proof, it suffices to prove that $G_{\a\b}$ has at most two orbits on $\Ga_2(\a)\cap\Ga_2(\b)$. To see this, we may assume that $G_{\a\b}$ is intransitive on $\Ga(\a)\cap\Ga_2(\b)$. Then there exist $\d_1, \d_2\in\Ga_2(\a)\cap\Ga_2(\b)$ such that the orbits $\d_1^{G_{\a\b}}\neq \d_2^{G_{\a\b}}$. Then $\d_1,\d_2$ are not adjacent to $\a$ and $\b$. Since $\Ga$ is $(G,3)$-set-homogeneous, there exists a $g\in G$ such that $\d_1^g=\d_2$, and $\a^g=\b$ and $\b^g=\a$. For any $\d\in \Ga_2(\a)\cap\Ga_2(\b)$, also since $\Ga$ is $(G,3)$-set-homogeneous, there exists $h\in G_{\{\a,\b\}}$ such that $\d^h=\d_1$. If $h\in G_{\a\b}$, then $\d\in \d_1^{G_{\a\b}}$. If $h\notin G_{\a\b}$, then $h$ swaps $\a$ and $\b$ and so $hg\in G_{\a\b}$. Furthermore, $\d^{hg}=\d_1^g=\d_2$, and hence $\d\in \d_2^{G_{\a\b}}$. Thus, $\Ga_2(\a)\cap\Ga_2(\b)=\d_1^{G_{\a\b}}\cup \d_2^{G_{\a\b}}$.
The proof is completed.
\hfill\qed
\end{proof}

The next lemma gives some basic properties about strongly regular graphs, see \cite[\textsection{1.1.1-1.1.2}]{BrouwerSRG}.

\begin{lem}\label{lem:drg-basic}
Let $\Ga$ be a strongly regular graph with parameters $(v, k, \lambda, \mu)$. Then the following hold:
\begin{enumerate}
  \item [{\rm (1)}]\ The complement $\overline{\Ga}$ of $\Ga$ is strongly regular with parameters $(v, \overline{k}, \overline{\ld}, \overline{\mu})$, where $\overline{k}=v-k-1$, $\overline{\ld}=v-2k+\mu-2$ and $\overline{\mu}=v-2k+\lambda$;
  \item [{\rm (2)}]\ $0\leq \ld\leq k-1$ and $0\leq\mu\leq k$;
  \item [{\rm (3)}]\ $k(k-1-\lambda)=\mu(v-1-k)$.
\end{enumerate}
\end{lem}

Let $\mathbf{P}_{3}$ be a path of length $2$. For a graph $\Ga$, let
\[
\begin{array}{l}
\Ome_1(\Ga):=\{U\subseteq V(\Ga) \mid [U]\cong \K_1\cup\K_2 \},\\
\Ome_2(\Ga):=\{U\subseteq V(\Ga) \mid [U]\cong \mathbf{P}_{3}\},\\
\Ome_3(\Ga):=\{U\subseteq V(\Ga) \mid [U]\cong \K_3\},\\
\Ome_4(\Ga):=\{U\subseteq V(\Ga) \mid [U]\cong 3\K_1\}.\\
\end{array}

\begin{theorem}\label{th-main}
For each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.
Moreover, { if $\Ga$ is a $k$-homogeneous graph with $k\geq 2$, then} one of the following holds:
\begin{itemize}
\item [{\rm(1)}] $k\geqslant 5$, and $\Ga$ is one of { $\C_5 (\cong\overline{\C_5})$, $\K_3\square\K_3 (\cong \overline{\K_3\square\K_3})$, $n\K_m$, and $\K_{n[m]}$; furthermore, $\Ga$ is homogeneous.}

{ \item[(2)] $\Ga$ is either the Schl\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\Ga$ is $4$-homogeneous but not $5$-homogeneous.

\item[(3)] $\Ga$ is either $\K_n\square\K_n$ with $n\geqslant4$, or a graph from Theorem~\ref{th:3sethomo}(1)--(4), or the complement of any such graph; furthermore, $\Ga$ is $3$-homogeneous but not $4$-homogeneous.

\item[(4)] $\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\Ga$ is $2$-homogeneous but not $3$-homogeneous. }

\end{itemize}
\end{theorem}

\begin{itemize}
\item[{\rm(1)}] the affine polar graph $\mathrm{VO}_{2m}^{\epsilon}(2)$ with $m\geq 1$ and $\epsilon \in \{+,-\}$;

\item[{\rm(2)}]  the elliptic orthogonal graph $\mathrm{\Gamma}(\mathrm{O}^-_6(q))$ with $q$ a prime power;

\item[{\rm(3)}] an orbital graph of the Higman-Sims group $\mathrm{HS}$ acting on $100$ points;

\item[{\rm(4)}] an orbital graph of the McLaughlin group $\mathrm{McL}$ acting on $275$ points.
\end{itemize}
In particular, all $3$-set-homogeneous graphs are $3$-homogeneous.
\end{theorem}

Note that all graphs in Theorem~\ref{th:3sethomo}~(1)--(4) are actually $3$-homogeneous graphs, which {are known} in the literature.

The rest of the paper is organised as follows. In Sections~\ref{subsec:2reduce}--\ref{sec6:proof-main-theorems}, we deal with finite $(G, 2)$-set-homogeneous graphs.
In Section~\ref{sec6:proof-main-theorems}, we prove Theorem~\ref{th:2sethomo} and Theorem~\ref{trans-rank-4}.
In Section~\ref{set:3sethom}, we deal with finite $(G, 3)$-set-homogeneous graphs, and prove Theorem~\ref{th:3sethomo}.
In Section~\ref{set:proof-of-main}, we prove Theorem~\ref{th-main}.

\section{A reduction}\label{subsec:2reduce}
In this section, we shall reduce the proof of Theorem~\ref{th:2sethomo} to two cases: $G$ is a primitive permutation group of rank $5$ { and odd order}, and $G$ is a primitive permutation group of rank $4$ which has exactly one self-paired orbital. The former case will be treated in Section~\ref{subset:prime-rank5}, and the latter case will be treated in Sections~\ref{subsec:affineprimitive-rank4} and \ref{subsec:almostsimple-rank4}. Based on these, Theorem~\ref{th:2sethomo} and Theorem~\ref{trans-rank-4} will be proved in Section~\ref{sec6:proof-main-theorems}.

We first introduce several terms of permutation groups.
Let $\Ome$ be a finite nonempty set. Denote by $\Sym(\Ome)$ the symmetric group on $\Ome$. Let $G\leqslant\Sym(\Ome)$ be transitive.
Let $\Ome\times\Ome=\{(u,v)\mid u,v\in \Ome\}$.
An orbit $(u,v)^G$ of $G$ on $\Ome\times\Ome$ is called an {\it orbital} of $G$ on $\Ome$, and
$(v,u)^G$ is called the {\it paired orbital} of $(u,v)^G$.
An orbital  $(u,v)^G$ is called {\it self-paired} if it equals $(v,u)^G$.
The orbital $(u,u)^G$ is called {\it trivial}, and the others are non-trivial. To an orbital $E$ we
associate the digraph with vertex set $\Ome$ and arc set $E$, called the {\em orbital digraph} for
$E$, denoted by the pair $(\Ome, E)$. For a given point $u$, each orbital $E=(u,v)^G$ corresponds to an orbit of $G_u$ on $\Ome$, namely, $\{v\ |\ (u,v)\in E\}$. The orbits of $G_u$ on $\Om$ are called {\em suborbits} of $G$, and their lengths are called the {\em subdegrees} of $G$.
For a set $\Ome$, let $\Ome^{\{2\}}$ be the set of 2-subsets of $\Ome$, namely, $\Ome^{\{2\}}=\{\{u,v\}\mid u\not=v\in \Ome\}$,  and let $\Ome^{(2)}=\{(u,v)\mid u\not=v\in \Ome\}$.

\begin{lemma}\label{rank<=5}
Let $\Ga=(V,E)$ be a $(G, 2)$-set-homogeneous graph, where $G\leq\Aut(\Ga)$.
Then $G$ is a permutation group on $V$  of rank at most $5$, and
one of the following holds:
\begin{itemize}
\item[{\rm(1)}] $G$ is of rank $2$, and $\Ga$ is a complete graph or an empty graph.

\item[{\rm(2)}] $G$ is of rank $3$, and {either $\Ga$ is a complete graph or an empty graph, or $\Ga$ is $(G, 2)$-homogeneous}.

\item[{\rm(3)}] $G$ is of rank $4$, and exactly one of the $3$ non-trivial orbitals is self-paired.

\item[{\rm(4)}] $G$ is of rank $5$, and none of the $4$ non-trivial orbitals is self-paired.

\end{itemize}

\begin{theorem}\label{rank-4}
Let $G$ be an almost simple primitive rank $4$ permutation group on a set $\Om$ with a unique non-self-paired orbital.
Then $G=\PSU_3(3)$ with stabiliser $\PSL_3(2)$, and the subdegrees are $1$, $7$, $7$ and $21$. Moreover, $\mathbf{N}_{\mathrm{Sym}(\mathit{\Omega})}(G)=\mathrm{P\Gamma U}_3(3)$ is a primitive rank $3$ group with subdegrees $1$, $14$ and $21$.
\end{theorem}

The main purpose of this paper is to solve the long-standing open problem:

\vskip0.1in
\f{\bf Problem A.} {\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}
\vskip0.1in

Before proceeding, we give some background on this topic.
The strongest possible symmetry condition that one can impose {on a graph is {\it homogeneity}.}
A graph is called {\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},
and Gardiner~\cite{Gardiner1976} in the 1970s.
{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':
\[\C_5,\ \K_3\square\K_3,\ n\K_m,\ \K_{n[m]}.\]
A graph $\Ga$ is called {\em set-homogeneous} if isomorphic subgraphs of $\Ga$ are equivalent under $\Aut(\Ga)$.
{In 1978, Ronse~\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.
{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\cite{Macpherson-survey} for an overview.}

{One of these weaker conditions is as follows.}

\begin{definition}\label{Def-homog}
{\rm
Let $\Ga=(V,E)$ be a graph, $G\leqslant\Aut(\Ga)$, and  $k$ a positive integer.
\begin{itemize}
\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\Ga$ of order at most $k$
extends to an automorphism of $\Ga$ { contained} in $G$, then $\Ga$ is called a {\it $(G, k)$-homogeneous} graph.
(The {\it order} of a graph is the number of vertices.)

\item[(2)] If any two isomorphic induced subgraphs of $\Ga$ of order at most $k$ are equivalent under $G$, then $\Ga$ is called a {\it $(G,k)$-set-homogeneous} graph.

\item[(3)] {When $G=\Aut(\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\em $k$-homogeneous} or {\em $k$-set-homogeneous}, respectively.}
\end{itemize}
}
\end{definition}

A major theme in the area has been the following classification problem.

\vskip0.1in
\f{\bf Problem B.}
{\rm Classify $k$-homogeneous graphs and $k$-set-homogeneous graphs.}
\vskip0.1in

This paper completely solves the problem, and particularly shows that a finite graph is $k$-homogeneous if and only if it is $k$-set-homogeneous.

{The study of Problem~B has lasted for over half a century, dating back to Sheehan and Gardiner's classification of finite homogeneous graphs in the 1970s mentioned above.} Later, Lachlan and Woodrow~\cite{Lachlan-Woodrow} obtained a classification of infinite countable homogeneous graphs. For more results regarding homogeneous graphs, we refer the reader to \cite{Enomoto} and \cite{Gray,Gray-Macpherson,GMPR,HH,Henson}.

By definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.
It is infeasible to obtain { an explicit list of vertex-transitive graphs.}
However, { an explicit classification has been achieved for}
finite $k$-homogeneous graphs with $k\geqslant 2$ in a collection of papers. Cameron~\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;
Buczak~\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.
{We observe} that a graph $\Ga$ is $2$-homogeneous if and only if $\Ga$ and its complement $\ov\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.
Note that primitive permutation groups of rank $3$ have been classified in \cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}

The main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}
To state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\K_n$, the cycle with $n$ vertices by $\C_n$, the complete multipartite graph with $n$ parts of
size $m$ by $\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\K_m$) by $n\K_m$. Let $\K_n\square\K_n$ and $\K_n\times\K_n$ be the Cartesian product and direct product of two $\K_n$'s, respectively. For a graph $\Ga$, we denote by $\ov\Ga$ the {\it complement} of $\Ga$. Note that $\overline{\K_n\square\K_n}=\K_n\times\K_n$.

\vskip0.1in
\f{\bf Problem A.} {\rm {Classify the finite graphs in which, for a given positive integer $k$, any two isomorphic induced subgraphs of order at most $k$ are equivalent under the automorphism group.}}
\vskip0.1in

Before proceeding, we give some background on this topic.
The strongest possible symmetry condition that one can impose {on a graph is {\it homogeneity}.}
A graph is called {\em homogeneous} if every isomorphism between any pair of isomorphic induced subgraphs extends to an automorphism of the graph. The study of graphs with {certain homogeneity-type properties} was initiated by Sheehan~{\cite{Sheehan1972,Sheehan1973,Sheehan1973a,Sheehan1974}},
and Gardiner~\cite{Gardiner1976} in the 1970s.
{It turns out that} the only finite homogeneous graphs are the following `trivial graphs':
\[\C_5,\ \K_3\square\K_3,\ n\K_m,\ \K_{n[m]}.\]
A graph $\Ga$ is called {\em set-homogeneous} if isomorphic subgraphs of $\Ga$ are equivalent under $\Aut(\Ga)$.
{In 1978, Ronse~\cite{Ronse}} proved that every finite set-homogeneous graph is homogeneous.
{Slightly weaker versions of homogeneity have been studied in the literature} (see Definition~\ref{Def-homog} for instance) and also {have been extended} to other mathematical structures. Over the past 50 years, { a fair amount of work has been done on objects with certain homogeneity-type properties. The reader may refer to the excellent survey paper of Macpherson~\cite{Macpherson-survey} for an overview.}

\begin{definition}\label{Def-homog}
{\rm
Let $\Ga=(V,E)$ be a graph, $G\leqslant\Aut(\Ga)$, and $k$ a positive integer.
\begin{itemize}
\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\Ga$ of order at most $k$
extends to an automorphism of $\Ga$ { contained} in $G$, then $\Ga$ is called a {\it $(G, k)$-homogeneous} graph.
(The {\it order} of a graph is the number of vertices.)

\item[(3)] {When $G=\Aut(\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\em $k$-homogeneous} or {\em $k$-set-homogeneous}, respectively.}
\end{itemize}
}
\end{definition}

By definition, $1$-homogeneous graphs and $1$-set-homogeneous graphs are exactly vertex-transitive graphs.
It is infeasible to obtain { an explicit list of vertex-transitive graphs.}
However, { an explicit classification has been achieved for}
finite $k$-homogeneous graphs with $k\geqslant 2$ in a collection of papers. Cameron~\cite{Cameron-6-transitive} proved that the list of finite $5$-homogeneous graphs is the same as that for homogeneous graphs;
Buczak~\cite{Buczak} classified finite $4$-homogeneous graphs; and Cameron and Macpherson~\cite{Cameron-3-hom} classified finite $3$-homogeneous graphs.
{We observe} that a graph $\Ga$ is $2$-homogeneous if and only if $\Ga$ and its complement $\ov\Ga$ are both arc-transitive. {Thus, a finite $2$-homogeneous graph is a complete multipartite graph or its complement, or an orbital graph of a primitive permutation group of rank $3$.
Note that primitive permutation groups of rank $3$ have been classified in \cite{Bannai,Kantor,Liebeck-affine,Liebeck-Saxl}.}

The main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}
To state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\K_n$, the cycle with $n$ vertices by $\C_n$, the complete multipartite graph with $n$ parts of
size $m$ by $\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\K_m$) by $n\K_m$. Let $\K_n\square\K_n$ and $\K_n\times\K_n$ be the Cartesian product and direct product of two $\K_n$'s, respectively. For a graph $\Ga$, we denote by $\ov\Ga$ the {\it complement} of $\Ga$. Note that $\overline{\K_n\square\K_n}=\K_n\times\K_n$.

\begin{definition}\label{Def-homog}
{\rm
Let $\Ga=(V,E)$ be a graph, $G\leqslant\Aut(\Ga)$, and  $k$ a positive integer.
\begin{itemize}
\item[(1)] If each isomorphism between any pair of isomorphic induced subgraphs of $\Ga$ of order at most $k$
extends to an automorphism of $\Ga$ { contained} in $G$, then $\Ga$ is called a {\it $(G, k)$-homogeneous} graph.
(The {\it order} of a graph is the number of vertices.)

\item[(2)] If any two isomorphic induced subgraphs of $\Ga$ of order at most $k$ are equivalent under $G$, then $\Ga$ is called a {\it $(G,k)$-set-homogeneous} graph.

\item[(3)] {When $G=\Aut(\Ga)$, a $(G, k)$-homogeneous or $(G, k)$-set-homogeneous graph is simply called {\em $k$-homogeneous} or {\em $k$-set-homogeneous}, respectively.}
\end{itemize}
}
\end{definition}

\begin{theorem}\label{th:3sethomo}
Let $\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\Ga$ is isomorphic either to one of the graphs: $\C_5$, $\K_n$, $\K_{m[b]}$, $\K_n\square\K_n$, or to one of the following graphs:
\begin{itemize}
\item[{\rm(1)}] the affine polar graph $\mathrm{VO}_{2m}^{\epsilon}(2)$ with $m\geq 1$ and $\epsilon \in \{+,-\}$;

\item[{\rm(2)}]  the elliptic orthogonal graph $\mathrm{\Gamma}(\mathrm{O}^-_6(q))$ with $q$ a prime power;

\item[{\rm(3)}] an orbital graph of the Higman-Sims group $\mathrm{HS}$ acting on $100$ points;

\item[{\rm(4)}] an orbital graph of the McLaughlin group $\mathrm{McL}$ acting on $275$ points.
\end{itemize}
In particular, all $3$-set-homogeneous graphs are $3$-homogeneous.
\end{theorem}

\begin{abstract}
A classification is given of finite $k$-set-homogeneous graphs for $k\geqslant 2$, leading to a striking result that each finite $k$-set-homogeneous graph is $k$-homogeneous. It shows that $3$-set-homogeneous graphs are rare, consisting of the following graphs and their complements: $\C_5$, $\K_n\square\K_n$, $n\K_m$, the Schl\"{a}fli graph of order 27, the Higman-Sims graph, the MaLaughlin graph, {affine polar graphs, and elliptic orthogonal graphs}. As an ingredient for the proof, it is shown that all orbitals in a primitive permutation group of rank $4$ are self-paired, except for $\PSU_3(3)$ acting on 36 points.
\bigskip

The main purpose of this paper is to solve Problem~B by proving that every $k$-set-homogeneous graph is $k$-homogeneous for $k\geqslant 2$, { so that $k$-set-homogeneous graphs are explicitly classified, as described in the previous paragraph.}
To state {our main theorem}, we fix some graph notation. Let $m$ and $n$ be positive integers. We denote the complete graph with $n$ vertices by $\K_n$, the cycle with $n$ vertices by $\C_n$, the complete multipartite graph with $n$ parts of
size $m$ by $\K_{n[m]}$, and its complement (the disjoint union of $n$ copies of $\K_m$) by $n\K_m$. Let $\K_n\square\K_n$ and $\K_n\times\K_n$ be the Cartesian product and direct product of two $\K_n$'s, respectively. For a graph $\Ga$, we denote by $\ov\Ga$ the {\it complement} of $\Ga$. Note that $\overline{\K_n\square\K_n}=\K_n\times\K_n$.

{\noindent{\bf Remark on Theorem~\ref{th-main}.}\ For $k\geq2$, all $k$-homogeneous graphs have already been classified in the literature, { as mentioned above}, and so all graphs in Theorem~\ref{th-main}~(1)--(4) are known.}\medskip

\begin{theorem}\label{th:2sethomo}
Let $\Ga=(V,E)$ be a finite $(G,2)$-set-homogeneous graph, where $G\leq\Aut(\Ga)$.
Then one of the following holds:
\begin{itemize}
\item[{\rm(1)}] $\Ga\cong\K_n$ or $n\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.

\begin{theorem}\label{th:3sethomo}
Let $\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\Ga$ is isomorphic either to one of the graphs: $\C_5$, $\K_n$, $\K_{m[b]}$, $\K_n\square\K_n$, or to one of the following graphs:
\begin{itemize}
\item[{\rm(1)}] the affine polar graph $\mathrm{VO}_{2m}^{\epsilon}(2)$ with $m\geq 1$ and $\epsilon \in \{+,-\}$;

\begin{theorem}[\cite{Liebeck-affine}]\label{subdegree}
Let $X$ be a finite primitive affine permutation group of rank $3$ and of degree $p^d$, with socle $V$, where $V\cong\mz_p^d$ for some prime $p$, and let $X_0$ be the stabiliser of the zero vector in $V$. Then $X_0$ belongs to one of the following classes $($and, conversely, each of the possibilities listed below does give rise to a rank $3$ affine group$)$.
\begin{itemize}
\item[\rm (A)] Infinite classes. These are
\begin{itemize}
\item[\rm (1)] $X_0\leq \mathrm{\Gamma L}_1(p^d)$: all possibilities for $X_0$ are determined in~\cite[\textsection{3}]{Foulser-Kallaher-solvable-rank-3};
\item[\rm (2)] $X_0$ imprimitive: $X_0$ stabilises a pair $\{V_1,V_2 \}$ of subspaces of $V$ where $V=V_1 \oplus V_2$ and $\dim(V_1)=\dim(V_2)$; moreover, $(X_0)_{V_i}$ is transitive on $V_i\setminus 0$ for $i=1$, $2$ (and hence $ X_0$ is determined by Hering's Theorem~\cite{Hering});
\item[\rm (3)] tensor product case: for some $a$, $q$ with $q^a= p^d$, consider $V$ as a vector space $\GF(q)^a$ of dimension $a$ over $\GF(q)$; then $X_0$ stabilises a decomposition of $\GF(q)^a$ as a tensor product $V_1 \otimes V_2$ where $\dim_{\GF(q)}(V_1)=2$; moreover, $(X_0)^{V_2}\vartriangleright \SL(V_2)$, or $(X_0)^{V_2}=\A_7<\SL_4(2)$ $($and $p=q=2$, $d=a=8)$, or $\dim_{\GF(q)}(V_2)\leq 3$;
\item[\rm (4)] $X_0\vartriangleright \SL_a(q) $ and $p^d=q^{2a}$;
\item[\rm (5)] $X_0\vartriangleright \SL_2(q) $ and $p^d=q^{6}$;
\item[\rm (6)] $X_0\vartriangleright \mathrm{SU}_a(q) $ and $p^d=q^{2a}$;
\item[\rm (7)] $X_0\vartriangleright \Omega^{\pm}_{2a}(q)$ $($and if $q$ is odd, $X_0$ contains an automorphism interchanging the two orbits of $\Omega^{\pm}_{2a}(q)$ on nonsingular $1$-spaces$)$;
\item[\rm (8)] $X_0\vartriangleright \SL_5(q) $ and $p^d=q^{10}$ $($from the action of $\SL_5(q)$ on $\bigwedge^2( \GF(q) ^5)$, the exterior square of $\GF(q)^5)$;

\begin{lemma}\label{lem:class:b-c}
If $G$ satisfies Theorem~{\rm\ref{subdegree}} {\rm(B)--(C)}, then one of the following holds:
\begin{enumerate} [\rm (1)]
\item $\Ga\cong \mathrm{VO}^{-}_6(2)$ or $\overline{\mathrm{VO}^{-}_6(2)}$, arising from the case where $p^d=2^6$ and $G_0\leq \N_{\GL_6(2)}(3^{1+2})$ in {\rm(B)};
\item $\Ga\cong \K_9\square\K_9$ or $\K_9\times\K_9$, arising from the case where $p^d=3^4$ and $G_0\leq \N_{\GL_4(3)}(2^{1+4}) $ in {\rm(B)};
\item $\Ga\cong \mathrm{VO}^{+}_8(2)$ or $\overline{\mathrm{VO}^{+}_8(2)}$, arising from the case where $p^d=2^8$ and $\soc(G_0/\Z(G_0))=\A_9<\mathrm{P\Omega}^{+}_8(2)$ in {\rm(C)}.
\end{enumerate}
\end{lemma}

\begin{proof} Assume that $G$ is of type {\rm(A1)}. Then $G\leq \mathrm{A\Gamma L}_1(p^d)$. Let $\{\alpha, \beta\}$ be an edge of $\Ga$.
Then $|V(\Ga)|=p^d$ and $G_{\alpha\beta}\leq \ZZ_d$. Furthermore, $G_{\a\b}$ is semiregular on $V(\Ga) \setminus \{\alpha, \beta\}$.
From Lemma~\ref{lm:Gab10orbits} it follows that $G_{\alpha\beta}$ has at most $12$ orbits on $V(\Ga) \setminus \{\alpha, \beta\}$.
Therefore, $G_{\alpha\beta}$ has one orbit of length at least $(|V(\Ga)|-2)/12$.
It follows that $|G_{\alpha\beta}|\geq (|V(\Ga)|-2)/12$, which means
\[ p^d-2 \leq 12d.\]
This implies that
\[
p^d\in\{2^i (1\leq i\leq6), 3^j (1\leq j\leq 3), 5^k, 7^k, 11^k (1\leq k\leq 2), 13\}.
\]
Since $\Ga$ has girth $3$ and diameter $2$ by Hypothesis~\ref{hyp}, it follows that $|V(\Ga)|=p^d\geq7$. For all of the remaining possible values of $p^d$, we make use of Magma~\cite{BCP} to construct all primitive rank $3$ subgroups $G$ of $\mathrm{A\Gamma L}_1(p^d)$, and then for each rank $3$ graph arising from $G$, we can further check if $|\Om_i(\Ga)|$ divides $|G|$ for $i=1,2,3,4$, where $\Om_i(\Ga)$ is given in Lemma~\ref{lm:Gaorder}. We find that the only possibility is $p^d=3^2$, and $\Ga\cong \mathrm{Paley}(9)\cong\K_3\square\K_3$. \hfill\qed
\end{proof}

{\noindent{\bf Remark on Theorem~\ref{th-main}.}\ For $k\geq2$, all $k$-homogeneous graphs have already been classified in the literature, { as mentioned above}, and so all graphs in Theorem~\ref{th-main}~(1)--(4) are known.}\medskip

By definition, a $k$-homogeneous graph is $k$-set-homogeneous.
The converse of this statement is clearly true for $k=1$.
{In the case where $k\ge4$,}
{Devillers~\cite{Devillers-2002paper}} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\"{a}fli graph on $27$ vertices and its complement, which are $4$-homogeneous (see, for example, \cite[Proposition~3.2.1]{Devillers}).
{So, to prove Theorem~\ref{th-main}, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and 3.}

Before stating the results, we introduce some families of graphs.

\begin{definition}\label{Def-graph-families}
{\rm \begin{itemize}
\item[\rm (1)]  Let ${\rm GF}(q)$ be a field of order a prime power $q$ such that $q\equiv 1\ (\mod 4)$, and let $\o$ be a fixed primitive { element} of ${\rm GF}(q)$.
Let $S=\{\o^2, \o^4, \ldots, \o^{q-3}, \o^{q-1}\}$. The {\it Paley graph} of order $q$, denoted by {\rm Paley}$(q)$, is the graph with vertex set ${\rm GF}(q)$ { where} two vertices $x,y$ are adjacent if and only if $x-y\in S$.

\item[\rm (2)]  For a positive integer $n$, the {\it Hamming graph}, denoted by $\mathrm{H}(2,n)$, is the graph with vertex set {$\{(i, j) \mid 1\leqslant i, j\leqslant n\}$}, and for every vertex $(i, j)$, its neighbour set is
\[
\{ (i,j')\mid j' \in \{1,2,\dots,n \}\setminus \{j\}\} \cup \{ (i',j)\mid i' \in \{1,2,\dots,n \}\setminus \{i\}\}.
\]
Note that $\mathrm{H}(2,n)$ is just the Cartesian product $\K_n\square\K_n$.

\item[\rm (3)]  For $\epsilon \in \{+,-\}$, a positive integer $m$ and a prime power $q$, the {\it affine polar graph}, denoted by $\mathrm{VO}_{2m}^{\epsilon}(q)$, is a graph with vertex set $\GF(q)^{2m}$ equipped with a nondegenerate quadratic form $Q$ of type $\epsilon$, and two vertices $\alpha$ and $\alpha'$ being adjacent if and only if $Q(\alpha-\alpha')=0$ (see \cite[\textsection{3.3.1}]{BrouwerSRG}). Since a nondegenerate $2$-dimensional orthogonal vector space of type $-$ contains no singular vector, it follows that $\mathrm{VO}_{2}^{-}(q)\cong q^2\K_1$. As remarked in~\cite[\textsection{3.3.1}]{BrouwerSRG}, we see that $\mathrm{VO}_{2}^{+}(q)\cong \K_q\square\K_q=\mathrm{H}(2,q)$.

\item[\rm (4)] For a prime power $q$, the {\it elliptic orthogonal graph}, denoted by $\mathrm{\Gamma}(\mathrm{O}^{-}_6(q))$ is a graph with vertex set consisting of  all singular $1$-subspaces of the orthogonal space on {$\GF(q)^{6}$ of type $-$},
    and two vertices $\alpha=\langle w \rangle$ and $\alpha'=\langle w' \rangle$ being adjacent if and only if $w\perp w'$ (see \cite[\textsection{2.6.3}]{BrouwerSRG}). The Schl\"{a}fli graph is the complement of the elliptic orthogonal graph $\mathrm{\Gamma}(\mathrm{O}^{-}_6(2))$ (see \cite[Table~11.8]{BrouwerSRG}).
\end{itemize}}
\end{definition}

We recall that a transitive permutation group $G$ on a set $\Ome$ is {\it $2$-homogeneous} if
all $2$-subsets of $\Ome$ are equivalent under $G$.
In the next theorem, a classification of finite $2$-set-homogeneous graphs is given.

\begin{theorem}\label{th:2sethomo}
Let $\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\leqslant\Aut(\Ga)$.
Then one of the following holds:
\begin{itemize}
\item[{\rm(1)}] $\Ga\cong\K_n$ or $n\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.

\item[{\rm(2)}] $\Ga\cong\K_{m[b]}$ or $m\K_b$, and  $G\leqslant X\wr Y$,
where $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.

\item[{\rm(3)}] $G$ is primitive of rank $3$, and $\Ga$ is $(G,2)$-homogeneous.

\item[{\rm(4)}] $G\cong\PSU_3(3)$ with vertex stabiliser $\PSL_3(2)$, $\Ga$ is of order $36$ and
valency $14$ or $21$, but $\Aut(\Ga)=\PSU_3(3).2$ is $2$-homogeneous on $\Ga$.

\item[(5)] $\Ga$ is a Paley graph of order $p^r$, where $p\equiv5 \pmod{8}$ and $r$ is odd, and
$G\leqslant\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\Ga$.
\end{itemize}
In particular, all $2$-set-homogeneous graphs are $2$-homogeneous.
\end{theorem}

{It is worth mentioning} that all graphs in Theorem~\ref{th:2sethomo}~(1)--(5) are $2$-homogeneous and already known in the literature.
{Since} a graph $\Ga$ is $(G, 2)$-set-homogeneous if and only if both $\Ga$ and $\ov\Ga$ are $G$-edge-transitive, { our} main approach for proving the above theorem is to {analyse} permutation groups of rank at most 5. It is well-known that all orbitals of a permutation group of rank 3 and {of even order} are self-paired. The following result shows that, except for a single group, all orbitals of a primitive permutation group of rank 4 are self-paired.

\begin{theorem}\label{trans-rank-4}
Let $G$ be a primitive permutation group of rank $4$. Then either all orbitals of $G$ are self-paired, or $G=\PSU_3(3)$ acting on $36$ points.
\end{theorem}

\begin{remark}
{\rm\begin{itemize}
\item[\rm (a)] More detailed information for the single group $\PSU_3(3)$ and its orbitals is given
in Theorem~\ref{rank-4}.
\item[\rm (b)] There are infinitely many imprimitive permutation groups of rank $4$ with exactly one non-trivial self-paired orbital.
Here is an example. Let $X$ be a $2$-homogeneous but not $2$-transitive permutation group, and let
$Y$ be a $2$-transitive permutation group. Then $X\wr Y$ is an imprimitive permutation group of rank $4$ with exactly one non-trivial self-paired orbital.
\end{itemize}}
\end{remark}

Our last theorem gives a classification of finite $3$-set-homogeneous graphs.

\begin{theorem}\label{th:3sethomo}
Let $\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\Ga$ is isomorphic either to one of the graphs: $\C_5$, $\K_n$, $\K_{m[b]}$, $\K_n\square\K_n$, or to one of the following graphs:
\begin{itemize}
\item[{\rm(1)}] the affine polar graph $\mathrm{VO}_{2m}^{\epsilon}(2)$ with $m\geqslant 1$ and $\epsilon \in \{+,-\}$;

\item[{\rm(2)}]  the elliptic orthogonal graph $\mathrm{\Gamma}(\mathrm{O}^-_6(q))$ with $q$ a prime power;

\item[{\rm(3)}] an orbital graph of the Higman-Sims group $\mathrm{HS}$ acting on $100$ points;

\item[{\rm(4)}] an orbital graph of the McLaughlin group $\mathrm{McL}$ acting on $275$ points.
\end{itemize}
In particular, all $3$-set-homogeneous graphs are $3$-homogeneous.
\end{theorem}

Note that all graphs in Theorem~\ref{th:3sethomo}~(1)--(4) are actually $3$-homogeneous graphs, which {are known} in the literature.

The rest of the paper is organised as follows. In Sections~\ref{subsec:2reduce}--\ref{sec6:proof-main-theorems}, we deal with finite $(G, 2)$-set-homogeneous graphs.
In Section~\ref{sec6:proof-main-theorems}, we prove Theorem~\ref{th:2sethomo} and Theorem~\ref{trans-rank-4}.
In Section~\ref{set:3sethom}, we deal with finite $(G, 3)$-set-homogeneous graphs, and prove Theorem~\ref{th:3sethomo}.
In Section~\ref{set:proof-of-main}, we prove Theorem~\ref{th-main}.

By definition, a $k$-homogeneous graph is $k$-set-homogeneous. For $k=1$ the converse is clearly true. For $k\ge4$, Devillers~\cite{Devillers-2002paper} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\"{a}fli graph on $27$ vertices and its complement (which are $4$-homogeneous). Hence, to prove Theorem~\ref{th-main}, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and $3$.

This is done by classifying finite $2$-set-homogeneous graphs (Theorem~\ref{th:2sethomo}, yielding in particular that all $2$-set-homogeneous graphs are $2$-homogeneous) and finite $3$-set-homogeneous graphs (Theorem~\ref{th:3sethomo}, yielding in particular that all $3$-set-homogeneous graphs are $3$-homogeneous), and then concluding Theorem~\ref{th-main} in Section~\ref{set:proof-of-main}.

By definition, a $k$-homogeneous graph is $k$-set-homogeneous. For $k=1$ the converse is clearly true. For $k\ge4$, Devillers~\cite{Devillers-2002paper} proved that every finite $5$-set-homogeneous graph is set-homogeneous, and that a $4$-set-homogeneous graph is either homogeneous or one of the Schl\"{a}fli graph on $27$ vertices and its complement (which are $4$-homogeneous). Hence, to prove the main theorem, it suffices to show that every $k$-set-homogeneous graph is also $k$-homogeneous for $k=2$ and $3$.

We first prove the following theorem.
\begin{theorem}\label{th:2sethomo}
Let $\Ga=(V,E)$ be a finite  $(G,2)$-set-homogeneous graph, where $G\leq\Aut(\Ga)$.
Then one of the following holds:
\begin{itemize}
\item[{\rm(1)}] $\Ga\cong\K_n$ or $n\K_1$, and $G$ is a $2$-homogeneous permutation group on $V$.

\item[{\rm(2)}] $\Ga\cong\K_{m[b]}$ or $m\K_b$, and  $G\leqslant X\wr Y$,
where $X,Y$ are $2$-homogeneous permutation groups of degree $b,m$, respectively.

\item[{\rm(3)}] $G$ is primitive of rank $3$, and $\Ga$ is $(G,2)$-homogeneous.

\item[{\rm(4)}] $G\cong\PSU_3(3)$ with vertex stabiliser $\PSL_3(2)$, $\Ga$ is of order $36$ and
valency $14$ or $21$, but $\Aut(\Ga)=\PSU_3(3).2$ is $2$-homogeneous on $\Ga$.

\item[(5)] $\Ga$ is a Paley graph of order $p^r$, where $p\equiv5 \pmod{8}$ and $r$ is odd, and
$G\leq\AGammaL_1(p^r)$ is $2$-set-homogeneous on $\Ga$.
\end{itemize}
In particular, all $2$-set-homogeneous graphs are $2$-homogeneous.
\end{theorem}

We next prove the following theorem.
\begin{theorem}\label{th:3sethomo}
Let $\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\Ga$ is isomorphic either to one of the graphs: $\C_5$, $\K_n$, $\K_{m[b]}$, $\K_n\square\K_n$, or to one of the following graphs:
\begin{itemize}
\item[{\rm(1)}] the affine polar graph $\mathrm{VO}_{2m}^{\epsilon}(2)$ with $m\geq 1$ and $\epsilon \in \{+,-\}$;

\item[{\rm(2)}]  the elliptic orthogonal graph $\mathrm{\Gamma}(\mathrm{O}^-_6(q))$ with $q$ a prime power;

\item[{\rm(3)}] an orbital graph of the Higman-Sims group $\mathrm{HS}$ acting on $100$ points;

\item[{\rm(4)}] an orbital graph of the McLaughlin group $\mathrm{McL}$ acting on $275$ points.
\end{itemize}
In particular, all $3$-set-homogeneous graphs are $3$-homogeneous.
\end{theorem}

Finally, we conclude the proof of the main theorem as follows:
\label{set:proof-of-main}

By definition, both 1-homogeneous graphs and 1-set-homogeneous graphs are exactly the vertex-transitive graphs. Assume now that $k\geq 2$. It is obvious by definition that e

\label{th-main}
For each positive integer $k$, a finite graph is $k$-set-homogeneous if and only if it is $k$-homogeneous.
Moreover, { if $\Ga$ is a $k$-homogeneous graph with $k\geqslant 2$, then} one of the following holds:
\begin{itemize}
\item [{\rm(1)}] $k\geqslant 5$, and $\Ga$ is one of { $\C_5 (\cong\overline{\C_5})$, $\K_3\square\K_3 (\cong \overline{\K_3\square\K_3})$, $n\K_m$, and $\K_{n[m]}$; furthermore, $\Ga$ is homogeneous.}

{ \item[(2)] $\Ga$ is either the Schl\"{a}fli graph on $27$ vertices or its complement (the two non-trivial orbital graphs of $\mathrm{PSU}_4(2)$ acting on the isotropic lines); furthermore, $\Ga$ is $4$-homogeneous but not $5$-homogeneous.

\item[(3)] $\Ga$ is either $\K_n\square\K_n$ with $n\geqslant4$, or a graph from the following theorem (items (1)--(4)), or the complement of any such graph; furthermore, $\Ga$ is $3$-homogeneous but not $4$-homogeneous.

We first prove the following theorem.
\begin{theorem}\label{th:3sethomo}
Let $\Ga=(V,E)$ be a finite $3$-set-homogeneous graph. Then, up to complement, $\Ga$ is isomorphic either to one of the graphs: $\C_5$, $\K_n$, $\K_{m[b]}$, $\K_n\square\K_n$, or to one of the following graphs:
\begin{itemize}
\item[{\rm(1)}] the affine polar graph $\mathrm{VO}_{2m}^{\epsilon}(2)$ with $m\geq 1$ and $\epsilon \in \{+,-\}$;

\item[{\rm(2)}]  the elliptic orthogonal graph $\mathrm{\Gamma}(\mathrm{O}^-_6(q))$ with $q$ a prime power;

\item[{\rm(3)}] an orbital graph of the Higman-Sims group $\mathrm{HS}$ acting on $100$ points;

\item[{\rm(4)}] an orbital graph of the McLaughlin group $\mathrm{McL}$ acting on $275$ points.
\end{itemize}
In particular, all $3$-set-homogeneous graphs are $3$-homogeneous.
\end{theorem}

\item[(4)] $\Ga$ is an orbital graph of a primitive permutation group of rank $3$ which is not covered by parts (1)--(3); furthermore, $\Ga$ is $2$-homogeneous but not $3$-homogeneous. }

\end{itemize}

Biconditional or Equivalence
Classification or Bijection

Question: Let be a finite graph, and let k be a positive integer. Call k-set-homogeneous if any two isomorphic induced subgraphs of with at most k vertices lie in the same orbit of Aut(). Call k-homogeneous if every isomorphism between any two isomorphic induced subgraphs of with at most k vertices extends to an automorphism of . Here nK_m denotes the disjoint union of n copies of K_m, K_{n[m]} the complete multipartite graph with n parts of size m, K_n\square K_n the Cartesian product of two copies of K_n, and an orbital graph is a graph whose edge set is an orbit of a permutation group on unordered pairs of vertices. Which statement gives the complete classification of finite k-set-homogeneous graphs?

For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if is k-homogeneous with k\ge 2, then one of the following holds: (1) k\ge 5, and is one of C_5, K_3\square K_3, nK_m, or K_{n[m]}; in fact is homogeneous. (2) is the Schl\"afli graph on 27 vertices or its complement; then is 4-homogeneous but not 5-homogeneous. (3) is K_n\square K_n for some n\ge 4, or its complement, or one of the affine polar graphs VO_{2m}^{\epsilon}(2) with m\ge 1 and \epsilon\in\{+,-\}, the elliptic orthogonal graphs \Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then is 3-homogeneous but not 4-homogeneous. (4) Otherwise, is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then is 2-homogeneous but not 3-homogeneous.

Choices:
B. For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \Gamma is k-homogeneous with k\ge 2, then one of the following holds: (1) k\ge 5, and \Gamma is one of C_5, K_3\square K_3, nK_m, or K_{n[m]}; in fact \Gamma is homogeneous. (2) \Gamma is the Schl\"afli graph on 27 vertices or its complement; then \Gamma is 4-homogeneous but not 5-homogeneous. (3) \Gamma is K_n\square K_n for some n\ge 3, or its complement, or one of the affine polar graphs VO_{2m}^{\epsilon}(2) with m\ge 1 and \epsilon\in\{+,-\}, the elliptic orthogonal graphs \Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \Gamma is 2-homogeneous but not 3-homogeneous.
C. For every positive integer k, every finite k-homogeneous graph is k-set-homogeneous. Moreover, every finite k-set-homogeneous graph belongs to one of the four families listed in (1)-(4) of the classification for k-homogeneous graphs.
D. For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \Gamma is k-homogeneous with k\ge 2, then one of the following holds: (1) k\ge 5, and \Gamma is one of C_5, K_3\square K_3, nK_m, or K_{n[m]}; in fact \Gamma is homogeneous. (2) \Gamma is the Schl\"afli graph on 27 vertices or its complement; then \Gamma is 4-homogeneous but not 5-homogeneous. (3) \Gamma is K_n\square K_n for some n\ge 4, or its complement, or one of the affine polar graphs VO_{2m}^{\epsilon}(2) with m\ge 1 and \epsilon\in\{+,-\}, the elliptic orthogonal graphs \Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \Gamma is 3-homogeneous but not 5-homogeneous. (4) Otherwise, \Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \Gamma is 2-homogeneous but not 3-homogeneous.
E. For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \Gamma is k-homogeneous with k\ge 2, then one of the following holds: (1) k\ge 5, and \Gamma is one of C_5, K_3\square K_3, nK_m, or K_{n[m]}; in fact \Gamma is homogeneous. (2) \Gamma is the Schl\"afli graph on 27 vertices or its complement; then \Gamma is 4-homogeneous but not 5-homogeneous. (3) \Gamma is K_n\square K_n for some n\ge 4, or one of the affine polar graphs VO_{2m}^{\epsilon}(2) with m\ge 1 and \epsilon\in\{+,-\}, the elliptic orthogonal graphs \Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points; then \Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3), or the complement of such a graph; then \Gamma is 2-homogeneous but not 3-homogeneous.

threshold n\ge 4 for K_n\square K_n in the 3-homogeneous-but-not-4-homogeneous case

dropped the converse direction and the per-case sharp homogeneity level conclusions

sharp exclusion '3-homogeneous but not 4-homogeneous' replaced by 'not 5-homogeneous'

complement closure moved from part (3) to part (4), omitting complements in the 3-homogeneous family

The stem defines the notions involved but does not reveal the classification or explicitly signal the correct option. There is no direct answer leakage beyond standard terminology.

The item asks for an equivalent statement of a theorem, so it is not a pure restatement of the stem, but it is still fundamentally theorem-recognition rather than a genuinely independent conclusion drawn from new data.

The question mainly tests recall of a precise classification statement and the ability to detect small textual perturbations. It does not require substantive mathematical generation or multi-step reasoning from the definitions.

The distractors are plausible and well-targeted: one is a weaker true statement, while others alter thresholds, homogeneity levels, or complement-closure details. These reflect realistic mathematical failure modes.

A technically well-constructed recall/checking item with strong distractors and no leakage, but it has limited generative reasoning value and leans heavily on theorem memorization.


Question: Let be a finite graph, and let k be a positive integer. Call k-set-homogeneous if any two isomorphic induced subgraphs of with at most k vertices lie in the same orbit of Aut(). Call k-homogeneous if every isomorphism between any two isomorphic induced subgraphs of with at most k vertices extends to an automorphism of . Here nK_m denotes the disjoint union of n copies of K_m, K_{n[m]} the complete multipartite graph with n parts of size m, K_n\square K_n the Cartesian product of two copies of K_n, and an orbital graph is a graph whose edge set is an orbit of a permutation group on unordered pairs of vertices. Which statement gives the complete classification of finite k-set-homogeneous graphs?

Choices:
B. For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \Gamma is k-homogeneous with k\ge 2, then one of the following holds: (1) k\ge 5, and \Gamma is one of C_5, K_3\square K_3, nK_m, or K_{n[m]}; in fact \Gamma is homogeneous. (2) \Gamma is the Schl\"afli graph on 27 vertices or its complement; then \Gamma is 4-homogeneous but not 5-homogeneous. (3) \Gamma is K_n\square K_n for some n\ge 3, or its complement, or one of the affine polar graphs VO_{2m}^{\epsilon}(2) with m\ge 1 and \epsilon\in\{+,-\}, the elliptic orthogonal graphs \Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \Gamma is 2-homogeneous but not 3-homogeneous.
C. For every positive integer k, every finite k-homogeneous graph is k-set-homogeneous. Moreover, every finite k-set-homogeneous graph belongs to one of the four families listed in (1)-(4) of the classification for k-homogeneous graphs.
D. For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \Gamma is k-homogeneous with k\ge 2, then one of the following holds: (1) k\ge 5, and \Gamma is one of C_5, K_3\square K_3, nK_m, or K_{n[m]}; in fact \Gamma is homogeneous. (2) \Gamma is the Schl\"afli graph on 27 vertices or its complement; then \Gamma is 4-homogeneous but not 5-homogeneous. (3) \Gamma is K_n\square K_n for some n\ge 4, or its complement, or one of the affine polar graphs VO_{2m}^{\epsilon}(2) with m\ge 1 and \epsilon\in\{+,-\}, the elliptic orthogonal graphs \Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points, or the complement of one of these; then \Gamma is 3-homogeneous but not 5-homogeneous. (4) Otherwise, \Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3); then \Gamma is 2-homogeneous but not 3-homogeneous.
E. For every positive integer k, a finite graph is k-set-homogeneous if and only if it is k-homogeneous. Moreover, if \Gamma is k-homogeneous with k\ge 2, then one of the following holds: (1) k\ge 5, and \Gamma is one of C_5, K_3\square K_3, nK_m, or K_{n[m]}; in fact \Gamma is homogeneous. (2) \Gamma is the Schl\"afli graph on 27 vertices or its complement; then \Gamma is 4-homogeneous but not 5-homogeneous. (3) \Gamma is K_n\square K_n for some n\ge 4, or one of the affine polar graphs VO_{2m}^{\epsilon}(2) with m\ge 1 and \epsilon\in\{+,-\}, the elliptic orthogonal graphs \Gamma(O^-_6(q)) with q a prime power, an orbital graph of the Higman-Sims group HS on 100 points, or an orbital graph of the McLaughlin group McL on 275 points; then \Gamma is 3-homogeneous but not 4-homogeneous. (4) Otherwise, \Gamma is an orbital graph of a primitive permutation group of rank 3 not already covered by (1)-(3), or the complement of such a graph; then \Gamma is 2-homogeneous but not 3-homogeneous.

The stem does not reveal the correct option. It introduces definitions and notation, then asks for the classification; the correct answer is not explicitly or implicitly singled out.

The item mainly asks the test-taker to recognize the exact classification theorem statement. This is essentially a theorem-recall/restatement question rather than a problem requiring selection among genuinely competing mathematical conclusions.

There is some reasoning pressure because the distractors are subtle near-miss variants involving boundary cases, converse strength, and complement closure. However, the task is still primarily precise recall/comparison of theorem statements, not generative mathematical reasoning.

The distractors are strong: they are plausible, mathematically close to the correct theorem, and exploit realistic failure modes such as weakened converses, altered thresholds, and incorrect sharp homogeneity levels.

A solid theorem-recognition MCQ with excellent distractors and no answer leakage, but it is weak on tautology avoidance and only moderately tests genuine generative reasoning.


Let $(X^n,\omega_0)$ be a compact K\"ahler manifold, and let $\omega(t)$ be the solution of the normalized K\"ahler-Ricci flow
\begin{equation}\label{eq--KRF}
\left\{
                \begin{aligned}
                  &\frac{\partial}{\partial t}\omega(t)=-\mathrm{Ric}(\omega(t))-\omega(t),\\
                  &\omega(0)=\omega_0,
                \end{aligned}
              \right.
\end{equation}
starting at $\omega_0$. We are interested in the case when the solution $\omega(t)$ is {\em immortal}, i.e. it exists for all $t\geqslant 0$. By a result of Tian-Zhang \cite{TiZh}, this happens if and only if the canonical bundle $K_X$ is nef, i.e. $c_1(K_X)$ is a limit of K\"ahler classes. We are interested in the behavior of the metrics $\omega(t)$ as $t\to\infty$, and refer to the second-named author's survey on this topic \cite{To}.

The Abundance Conjecture in birational geometry, and its natural extension to K\"ahler manifolds, predicts that if $K_X$ is nef then $K_X$ is semiample, which means that $\ell K_X$ is base-point free for some $\ell\geqslant 1$. Abundance is currently known for $n\leqslant 3$, and we will assume from now on that $K_X$ is indeed semiample.

In this case, the Kodaira dimension $m:=\kappa(X)$ satisfies $0\leqslant m\leqslant n$, and the linear system $|\ell K_X|$ for $\ell$ sufficiently divisible gives a fiber space (the Iitaka fibration of $X$)
\begin{equation}
f:X\to Y\subset\mathbb{P}^N \text{ with } \ell K_X=f^*(\mathcal{O}_{\mathbb{P}^N}(1)|_Y),
\end{equation}
 onto a normal projective variety $Y$ of dimension $\dim Y=m$. This implies that the generic fibers of $f$ are Calabi-Yau $(n-m)$-folds. The variety $Y=\mathrm{Proj}\oplus_{k\geqslant 0}H^0(X,kK_X)$ is the canonical model of $X$.

We define $D\subset Y$ as the closed subvariety given by the union of the singularities of $Y$ together with the discriminant locus of $f$, so that if we call $X^\circ:=X\backslash f^{-1}(D)$ and $Y^\circ:=Y\backslash D$, then $f:X^\circ\to Y^\circ$ is a proper holomorphic submersion with connected $(n-m)$-dimensional Calabi-Yau fibers.

In the case when $m=0$, we have that $Y$ is a point, $D=\emptyset$, $X$ itself is Calabi-Yau, and the behavior of the flow has been fully understood since the 1980s \cite{Cao}: the flow \eqref{eq--KRF} shrinks the metric smoothly to zero as $t\to\infty$ (in particular, $(X,\omega(t))$ converges to a point in the Gromov-Hausdorff topology), and reparametrizing the flow to have constant volume, it converges smoothly to a Ricci-flat K\"ahler metric on $X$.

The case $m=n$ is also special, because in this case the flow is volume non-collapsed, in the sense that $\mathrm{Vol}(X,\omega(t))\geqslant c>0$ for all $t\geqslant 0$. In this case $K_X$ is nef and big, and $X$ is known as a minimal model of general type. In this case, Tsuji \cite{Ts} and Tian-Zhang \cite{TiZh} proved that the flow converges smoothly on compact subsets of $X^\circ$ to a negative K\"ahler-Einstein metric $\omega_{\rm can}$ on $Y^\circ$ (pulled back via $f:X^\circ\to Y^\circ,$ which is an isomorphism in this case).

The remaining cases when $0<m<n$ turned out to be vastly more complicated, due to the fact that the flow is now volume collapsing as $t\to\infty$. This setup was first considered by Song-Tian in \cite{ST0,ST}. In this case, after much work by many people (see \cite{To} for references), it was recently shown by Hein and the first and second-named authors \cite{HLT} that as $t\to\infty$, we have $\omega(t)\to f^*\omega_{\rm can}$ smoothly on $X^\circ$, where $\omega_{\can}$ is a twisted K\"ahler-Einstein metric on $Y^\circ$, constructed by Song-Tian \cite{ST}.

In this paper we are interested in the global limiting behavior of $(X,\omega(t))$ as $t\to\infty$, in the Gromov-Hausdorff topology. The following basic conjecture is due to Song-Tian \cite[P.651]{ST0}, \cite[Conjectures 6.2 and 6.3]{ST3}, and it fits into their picture of analytic minimal model program:
\begin{conjecture}\label{con}
Let $(X^n,\omega_0)$ be a compact K\"ahler manifold with $K_X$ semiample, and let $\omega(t)$ be the solution of the K\"ahler-Ricci flow \eqref{eq--KRF}. Then as $t\to\infty$, $(X,\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\circ,\omega_{\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.
\end{conjecture}

This conjecture has received much attention since it was first posed in 2006. As mentioned above, we now know \cite{HLT} that $\omega(t)$ converges locally smoothly to $f^*\omega_{\rm can}$ on $X^\circ$. However, obtaining reasonable estimates for the flow near $f^{-1}(D)=X\backslash X^\circ$ has proved to be very challenging.

The case $m=n$ was studied first, and the conjecture is known in low dimensions by Guo-Song-Weinkove \cite{GSW} and Tian-Zhang \cite{TiZ} or under the additional assumption of a uniform Ricci lower bound by Guo \cite{Gu}. When $m=n$, Conjecture \ref{con} was settled by Wang \cite{Wang}. The harder case when $0<m<n$ has also been much studied. In particular, Conjecture \ref{con} was proved by Song-Tian-Zhang \cite{STZ} when $m=1$ and the generic fibers of $f$ are tori, and by Li and the second-named author \cite{LT} in arbitrary dimensions assuming that $Y$ is smooth and the divisorial components of $D$ have simple normal crossings (which always holds when $m=1$).

Let us remark that the uniform diameter bound and existence of subsequential Gromov-Hausdorff limits of $(X,\omega(t))$ was only achieved recently in \cite{JS}, and even more generally in \cite{GPSS, GPSS2,guedjT,vu2024} assuming only that $K_X$ is nef. Lastly, Sz\'ekelyhidi \cite{Sz} has very recently shown that in the setting of Conjecture \ref{con}, the metric completion of $(Y^\circ,\omega_{\rm can})$ is homeomorphic to $Y$, is a non-collapsed $\mathrm{RCD}(-1,2m)$-space, and inside this metric space, $Y\backslash Y^\circ$ has real Hausdorff codimension at least $2$. We will denote this metric completion by $(Y,d_{\rm can})$.

Our main result finally resolves Conjecture \ref{con} in general:

Let $(X^n,\omega_0)$ be a compact K\"ahler manifold, and let $\omega(t)$ be the solution of the normalized K\"ahler-Ricci flow
\begin{equation}\label{eq--KRF}
\left\{
 \begin{aligned}
 &\frac{\partial}{\partial t}\omega(t)=-\mathrm{Ric}(\omega(t))-\omega(t),\\
 &\omega(0)=\omega_0,
 \end{aligned}
 \right.
\end{equation}
starting at $\omega_0$. We are interested in the case when the solution $\omega(t)$ is {\em immortal}, i.e. it exists for all $t\geqslant 0$. By a result of Tian-Zhang \cite{TiZh}, this happens if and only if the canonical bundle $K_X$ is nef, i.e. $c_1(K_X)$ is a limit of K\"ahler classes. We are interested in the behavior of the metrics $\omega(t)$ as $t\to\infty$, and refer to the second-named author's survey on this topic \cite{To}.

In the case when $m=0$, we have that $Y$ is a point, $D=\emptyset$, $X$ itself is Calabi-Yau, and the behavior of the flow has been fully understood since the 1980s \cite{Cao}: the flow \eqref{eq--KRF} shrinks the metric smoothly to zero as $t\to\infty$ (in particular, $(X,\omega(t))$ converges to a point in the Gromov-Hausdorff topology), and reparametrizing the flow to have constant volume, it converges smoothly to a Ricci-flat K\"ahler metric on $X$.

The case $m=n$ is also special, because in this case the flow is volume non-collapsed, in the sense that $\mathrm{Vol}(X,\omega(t))\geqslant c>0$ for all $t\geqslant 0$. In this case $K_X$ is nef and big, and $X$ is known as a minimal model of general type. In this case, Tsuji \cite{Ts} and Tian-Zhang \cite{TiZh} proved that the flow converges smoothly on compact subsets of $X^\circ$ to a negative K\"ahler-Einstein metric $\omega_{\rm can}$ on $Y^\circ$ (pulled back via $f:X^\circ\to Y^\circ,$ which is an isomorphism in this case).

In this paper we are interested in the global limiting behavior of $(X,\omega(t))$ as $t\to\infty$, in the Gromov-Hausdorff topology. The following basic conjecture is due to Song-Tian \cite[P.651]{ST0}, \cite[Conjectures 6.2 and 6.3]{ST3}, and it fits into their picture of analytic minimal model program:
\begin{conjecture}\label{con}
Let $(X^n,\omega_0)$ be a compact K\"ahler manifold with $K_X$ semiample, and let $\omega(t)$ be the solution of the K\"ahler-Ricci flow \eqref{eq--KRF}. Then as $t\to\infty$, $(X,\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\circ,\omega_{\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.
\end{conjecture}

Let us remark that the uniform diameter bound and existence of subsequential Gromov-Hausdorff limits of $(X,\omega(t))$ was only achieved recently in \cite{JS}, and even more generally in \cite{GPSS, GPSS2,guedjT,vu2024} assuming only that $K_X$ is nef. Lastly, Sz\'ekelyhidi \cite{Sz} has very recently shown that in the setting of Conjecture \ref{con}, the metric completion of $(Y^\circ,\omega_{\rm can})$ is homeomorphic to $Y$, is a non-collapsed $\mathrm{RCD}(-1,2m)$-space, and inside this metric space, $Y\backslash Y^\circ$ has real Hausdorff codimension at least $2$. We will denote this metric completion by $(Y,d_{\rm can})$.

Our main result finally resolves Conjecture \ref{con} in general:

\begin{equation}\label{eq--KRF}
\left\{
                \begin{aligned}
                  &\frac{\de}{\de t}\omega(t)=-\Ric(\omega(t))-\omega(t),\\
                  &\omega(0)=\omega_0,
                \end{aligned}
              \right.
\end{equation}

\section{Introduction}
Let $(X^n,\omega_0)$ be a compact K\"ahler manifold, and let $\omega(t)$ be the solution of the normalized K\"ahler-Ricci flow
\begin{equation}\label{eq--KRF}
\left\{
 \begin{aligned}
 &\frac{\de}{\de t}\omega(t)=-\Ric(\omega(t))-\omega(t),\\
 &\omega(0)=\omega_0,
 \end{aligned}
 \right.
\end{equation}
starting at $\omega_0$. We are interested in the case when the solution $\omega(t)$ is {\em immortal}, i.e. it exists for all $t\geq 0$. By a result of Tian-Zhang \cite{TiZh}, this happens if and only if the canonical bundle $K_X$ is nef, i.e. $c_1(K_X)$ is a limit of K\"ahler classes. We are interested in the behavior of the metrics $\omega(t)$ as $t\to\infty$, and refer to the second-named author's survey on this topic \cite{To}.

In the case when $m=0$, we have that $Y$ is a point, $D=\emptyset$, $X$ itself is Calabi-Yau, and the behavior of the flow has been fully understood since the 1980s \cite{Cao}: the flow \eqref{eq--KRF} shrinks the metric smoothly to zero as $t\to\infty$ (in particular, $(X,\omega(t))$ converges to a point in the Gromov-Hausdorff topology), and reparametrizing the flow to have constant volume, it converges smoothly to a Ricci-flat K\"ahler metric on $X$.

In this paper we are interested in the global limiting behavior of $(X,\omega(t))$ as $t\to\infty$, in the Gromov-Hausdorff topology. The following basic conjecture is due to Song-Tian \cite[P.651]{ST0}, \cite[Conjectures 6.2 and 6.3]{ST3}, and it fits into their picture of analytic minimal model program:
\begin{conjecture}\label{con}
Let $(X^n,\omega_0)$ be a compact K\"ahler manifold with $K_X$ semiample, and let $\omega(t)$ be the solution of the K\"ahler-Ricci flow \eqref{eq--KRF}. Then as $t\to\infty$, $(X,\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\circ,\omega_{\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.
\end{conjecture}

Our main result finally resolves Conjecture \ref{con} in general:

In particular, in the case when $m=n$, our arguments are different from those in \cite{Wang}, so we obtain a new proof of that case. For general $m$, our proof follows a strategy introduced in \cite{LT}, where a key input is Perelman's monotonicity of the reduced volume, viewed as a parabolic analogue of the Bishop-Gromov volume comparison for Riemannian metrics with Ricci curvature bounded below. This allows for some control over how much time minimizing $\mathcal{L}$-geodesics spend in a neighborhood of the ``bad region'' $X\backslash X^\circ=f^{-1}(D)$. However, to push this strategy to the end, \cite{LT} had to use a result from \cite{GTZ2} which shows that when $Y$ is smooth and the divisorial components of $D$ have simple normal crossings, $\omega_{\rm can}$ is quasi-isometric to a {\em conical K\"ahler metric} near these components, up to a small logarithmic error.

\subsection{Reduction to an upper of $d_{\can}$ by $d_t$}In the following, let $d_t$ denote the distance function on $X$ defined by the metric $\omega(t)$. To prove Theorem \ref{main} we need to show that $(X,d_t)$ converges in the Gromov-Hausdorff topology to $(Y,d_{\rm can})$. Recall that we have the analytic subvariety $D\subset Y$ so that $Y^\circ=Y\backslash D$ is smooth and $f$ is a submersion over $Y^\circ$. Given $\ve>0$ we define
\begin{equation}
 V_{\epsilon}:=\{y\in Y\mid d_{\can}(D,y)<\epsilon\}, \quad \tilde V_{\epsilon}:=f^{-1}(V_{\epsilon}).
\end{equation}and let $(Y\setminus V_{\epsilon},d_{\can})$ denote the restriction of the metric $d_{\can}$ from $Y$ to $Y\setminus V_{\epsilon}$, and similarly let $(X\setminus \tilde V_{\epsilon},d_t)$ be restriction of $d_t$ from $X$ to $X\setminus \tilde V_{\epsilon}$.

\begin{lemma}\label{lem--dcan bound by dt}
 Suppose the following is true: given any small $\ve>0$ and two points $p, q\in X\setminus \tilde V_{\epsilon}$, we have
 \begin{equation}\label{eq--to prove}
 d_{\can}(f(p),f(q))\leq d_t(p,q)+\Psi(t^{-1}|\epsilon).
\end{equation}
In this case, we then have
\begin{equation}\label{agogna}
 \lim_{t\rightarrow\infty}d_{\gh}((X,d_t),(Y,d_{\can}))=0
\end{equation}
\end{lemma}
\begin{proof}
 Thanks to Lemma \ref{lem--gh close for smooth exhaustion}, to prove \eqref{agogna} is enough to show that
 \begin{equation}\label{eq--intermediate}
 d_{\gh}((X\setminus \tilde V_{\epsilon},d_t), (Y\setminus V_{\epsilon},d_{\can}))=\Psi(t^{-1}| \epsilon).
 \end{equation} To show this, we will show that the fibration map
 \begin{equation}
 f:X\setminus \tilde V_{\epsilon}\rightarrow Y\setminus V_{\epsilon}
 \end{equation}is the Gromov-Hausdorff approximation that we want.
Indeed, by Lemma \ref{lem--convexity on y}, and the $C^0_{\loc}$-convergence of $\omega(t)$ to $\omega_{\can}$ on $f^{-1}(Y^{\circ})$ in \eqref{conv}, we know that for any $p,q\in X\setminus \tilde V_{\epsilon}$, we have
 \begin{equation}
 d_t(p,q)\leq d_{\can}(f(p),f(q))+\Psi(t^{-1}|\epsilon).
 \end{equation}
Combining this with \eqref{eq--to prove} then gives
 \begin{equation}
 \left|d_t(p,q)-d_{\can}(f(p),f(q))\right|=\Psi(t^{-1}|\epsilon),
 \end{equation}
and since clearly $f$ is surjective, we obtain the desired Gromov-Hausdorff approximation, which shows \eqref{eq--intermediate}.
\end{proof}

Following the argument in \cite{LT}, we now show that Proposition \ref{prop--main to prove} implies Theorem \ref{main}:
\begin{proof}[Proof of Theorem \ref{main}, assuming Proposition \ref{prop--main to prove}]
Thanks to Lemma \ref{lem--dcan bound by dt}, it suffices to show that \eqref{eq--to prove} holds. For this, we first observe the following easy upper bound of the $\mathcal L$-distance. By Lemma \ref{lem--convexity on y}, given $p,q\in X\setminus \tilde V_{\epsilon}$, we can find a curve $\gamma$ contained in $Y\setminus V_{\epsilon'}$ connecting $f(p)$ and $f(q)$ such that
\begin{equation}
 \mathrm{length}_{\omega_{\can}}(\gamma)\leq d_{\can}(f(p),f(q))+\Psi(\epsilon'|\epsilon).
\end{equation}
Then using the fiber bundle structure of $f$ and the $C^0_{\loc}$ convergence of $\omega(t)$, arguing as in \cite[Proof of Lemma 9.1]{TWY2}, we can construct a curve $\tilde \gamma$ contained in $X^\circ$, connecting $p$ and $q$, such that
\begin{equation}\label{eq--length upper bound}
 \mathrm{length}_{\omega(T)}(\tilde \gamma)\leq d_{\can}(f(p),f(q))+\Psi(\epsilon'|\epsilon)+\Psi(T^{-1}|\epsilon').
\end{equation}Note that this curve $\tilde\gamma$ depends on the parameter $\epsilon'$ and for simplicity of notations, in the following, we omit the dependence on $\epsilon'$.

\begin{equation}\label{eq--KRF}
\left\{
                \begin{aligned}
                  &\frac{\de}{\de t}\omega(t)=-\Ric(\omega(t))-\omega(t),\\
                  &\omega(0)=\omega_0,
                \end{aligned}
              \right.
\end{equation}

\begin{theorem}\label{main}In the setting of Conjecture \ref{con}, we have that as $t\to\infty$ the flow $(X,\omega(t))$ converges in the Gromov-Hausdorff topology to $(Y,d_{\mathrm{can}})$. In particular, Conjecture \ref{con} holds.
\end{theorem}

In particular, in the case when $m=n$, our arguments are different from those in \cite{Wang}, so we obtain a new proof of that case. For general $m$, our proof follows a strategy introduced in \cite{LT}, where a key input is Perelman's monotonicity of the reduced volume, viewed as a parabolic analogue of the Bishop-Gromov volume comparison for Riemannian metrics with Ricci curvature bounded below. This allows for some control over how much time minimizing $\mathcal{L}$-geodesics spend in a neighborhood of the ``bad region'' $X\backslash X^\circ=f^{-1}(D)$. However, to push this strategy to the end, \cite{LT} had to use a result from \cite{GTZ2} which shows that when $Y$ is smooth and the divisorial components of $D$ have simple normal crossings, $\omega_{\rm can}$ is quasi-isometric to a {\em conical K\"ahler metric} near these components, up to a small logarithmic error.

Given that no such conical description is available for general $Y$, in this paper we refine the analysis in \cite{LT} to establish the existence of $\mathcal{L}$-geodesics that almost avoid a small neighborhood of $f^{-1}(D)$ (or more precisely, they visit this neighborhood a controlled number of times for a controlled length of time). After carefully choosing appropriate neighborhoods of $f^{-1}(D)$ (here the RCD property of $(Y,d_{\rm can})$ is crucial), and combining this almost-avoidance property with a H\"older estimate for $d_{\can}$, we complete the proof of Theorem \ref{main}.

The paper is organized as follows. In Section \ref{sec--pre}, we collect various results from the literature concerning estimates for $(X,\omega(t))$ and $(Y^{\circ},\omega_{\can})$. In Section \ref{sec-holder}, we establish a H\"older estimate for the potential of $\omega_{\can}$ following \cite{GKSS}, and then derive a H\"older estimate for $d_{\can}$ using \cite{Li}. In Section \ref{sec--reduction}, following \cite{LT}, we reduce the main theorem to an estimate asserting that the $d_{\can}$-distance can be bounded above by the $\mathcal{L}$-distance.
Section \ref{sec--main} contains the core of the paper. We first prove a general almost-avoidance principle, roughly stating that for suitably small neighborhoods of $f^{-1}(D)$, a generic $\mathcal{L}$-geodesic cannot intersect such neighborhoods too many times. Then we carefully choose the small neighborhoods of $f^{-1}(D)$ to apply the almost-avoidance principle to, and we use the RCD property of $(Y,d_{\can})$ to establish an upper volume bound for these neighborhoods.
 Finally combining the almost-avoidance principle with the H\"older estimate obtained in Section \ref{sec-holder}, we complete the proof.

\subsection*{Acknowledgements} We are grateful to G\'abor Sz\'ekelyhidi for very useful discussions, and to Shouhei Honda, Wenshuai Jiang, Xiaochun Rong, and Song Sun for email communications. The third-named author thanks Yifan Guo for discussions about Poincar\'e inequalities on minimal surfaces.  The first-named author is supported by Hong Kong RGC grants No. 14300623 and No. 14304225, and an Asian Young Scientist Fellowship. The second-named author was partially supported by NSF grant DMS-2404599.

For general $m$, the proof of Theorem~\ref{main} follows the strategy of \cite{LT}, using "Perelman's monotonicity of the reduced volume" to control how much time minimizing $\mathcal{L}$-geodesics spend near the "bad region'' $X\backslash X^\circ=f^{-1}(D)$. Since no conical description of $\omega_{\rm can}$ is available for general $Y$, the paper "refine[s] the analysis in \cite{LT} to establish the existence of $\mathcal{L}$-geodesics that almost avoid a small neighborhood of $f^{-1}(D)$ (or more precisely, they visit this neighborhood a controlled number of times for a controlled length of time)." After "carefully choosing appropriate neighborhoods of $f^{-1}(D)$ (here the RCD property of $(Y,d_{\rm can})$ is crucial)," and combining this almost-avoidance property with a H\"older estimate for $d_{\can}$, the proof is completed.

More structurally: (i) establish a H\"older estimate for the potential of $\omega_{\can}$ and hence for $d_{\can}$; (ii) reduce the main theorem to an estimate that the $d_{\can}$-distance is bounded above by the $\mathcal{L}$-distance; (iii) prove an "almost-avoidance principle" that a generic $\mathcal{L}$-geodesic cannot intersect suitably small neighborhoods of $f^{-1}(D)$ too many times; (iv) choose neighborhoods and use the RCD property of $(Y,d_{\can})$ to get an upper volume bound for them; (v) combine the almost-avoidance principle with the H\"older estimate to complete the proof.

For general $m$, the proof of To prove the main theorem, … follows the strategy of \cite{LT}, using "Perelman's monotonicity of the reduced volume" to control how much time minimizing $\mathcal{L}$-geodesics spend near the "bad region'' $X\backslash X^\circ=f^{-1}(D)$. Since no conical description of $\omega_{\rm can}$ is available for general $Y$, the paper "refine[s] the analysis in \cite{LT} to establish the existence of $\mathcal{L}$-geodesics that almost avoid a small neighborhood of $f^{-1}(D)$ (or more precisely, they visit this neighborhood a controlled number of times for a controlled length of time)." After "carefully choosing appropriate neighborhoods of $f^{-1}(D)$ (here the RCD property of $(Y,d_{\rm can})$ is crucial)," and combining this almost-avoidance property with a H\"older estimate for $d_{\can}$, the proof is completed.

More structurally: (i) establish a H\"older estimate for the potential of $\omega_{\can}$ and hence for $d_{\can}$; (ii) reduce the main theorem to an estimate that the $d_{\can}$-distance is bounded above by the $\mathcal{L}$-distance; (iii) prove an "almost-avoidance principle" that a generic $\mathcal{L}$-geodesic cannot intersect suitably small neighborhoods of $f^{-1}(D)$ too many times; (iv) choose neighborhoods and use the RCD property of $(Y,d_{\can})$ to get an upper volume bound for them; (v) combine the almost-avoidance principle with the H\"older estimate to complete the proof.,

\label{main}In the setting of \begin{conjecture}\label{con}
Let $(X^n,\omega_0)$ be a compact K\"ahler manifold with $K_X$ semiample, and let $\omega(t)$ be the solution of the K\"ahler-Ricci flow \eqref{eq--KRF}. Then as $t\to\infty$, $(X,\omega(t))$ converges in the Gromov-Hausdorff topology to the metric completion of $(Y^\circ,\omega_{\rm can})$. This is a compact metric space homeomorphic to the canonical model $Y$.
\end{conjecture} we have that as $t\to\infty$ the flow $(X,\omega(t))$ converges in the Gromov-Hausdorff topology to $(Y,d_{\mathrm{can}})$. In particular, the conjecture above holds.

Question:
Let $(X^n,\omega_0)$ be a compact K\"ahler manifold with semiample canonical bundle $K_X$, and let $\omega(t)$ be the solution of the normalized K\"ahler-Ricci flow
\[
\begin{cases}
\dfrac{\partial}{\partial t}\omega(t)=-\mathrm{Ric}(\omega(t))-\omega(t),\\
\omega(0)=\omega_0.
\end{cases}
\]
Let $Y$ be the canonical model of $X$, let $Y^\circ$ denote the regular part on which the canonical metric $\omega_{\rm can}$ is defined, and let $(Y,d_{\rm can})$ denote the metric completion of $(Y^\circ,\omega_{\rm can})$. Under these hypotheses, which statement about the long-time behavior of the flow is valid?

As $t\to\infty$, the metric spaces $(X,\omega(t))$ converge in the Gromov-Hausdorff topology to $(Y,d_{\rm can})$; equivalently, they converge to the metric completion of $(Y^\circ,\omega_{\rm can})$. Thus the predicted Gromov-Hausdorff limit description by the canonical model holds in this setting.

Choices:
B. As $t\to\infty$, the metric spaces $(X,\omega(t))$ converge in the Gromov-Hausdorff topology to $(Y^\circ,\omega_{\rm can})$ itself, without passing to its metric completion. Thus the singular set plays no role in the limiting metric space.
C. As $t\to\infty$, the metric spaces $(X,\omega(t))$ admit subsequences that converge in the Gromov-Hausdorff topology to $(Y,d_{\rm can})$. In particular, $(Y,d_{\rm can})$ occurs as a Gromov-Hausdorff limit of the flow.
D. As $t\to\infty$, the metric spaces $(X,\omega(t))$ converge in the Gromov-Hausdorff topology to a compact metric space homeomorphic to the canonical model $Y$, but the theorem does not identify this limit canonically with $(Y,d_{\rm can})$ or with the metric completion of $(Y^\circ,\omega_{\rm can})$.
E. As $t\to\infty$, the metric spaces $(X,\omega(t))$ converge in the Gromov-Hausdorff topology to $(Y,d_{\rm can})$, and moreover the convergence is smooth on all of $X$ after identifying $X$ with $Y$ through the canonical map.

The stem does not explicitly reveal the correct option. It introduces the relevant objects and asks for the valid asymptotic statement, but does not itself state full Gromov-Hausdorff convergence to the completed canonical metric space.

The item is essentially asking for the precise statement of a known theorem. The correct choice largely restates the theorem's conclusion rather than requiring transfer of the result to a new situation.

There is some reasoning pressure in separating full convergence from subsequential convergence, completion versus regular part, and GH convergence versus smooth convergence. However, the main task is still theorem recall, not substantial generative reasoning.

The distractors are plausible and mathematically meaningful: one is weaker-but-true sounding, one omits metric completion, one weakens canonical identification, and one overstates smoothness. These reflect common failure modes in reading geometric convergence results.

A solid recall-style theorem-identification MCQ with strong distractors and little answer leakage, but it is fairly tautological and only moderately tests genuine reasoning.