\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


Question: Let $k$ be a positive integer and let $\Gamma$ be a finite graph. Say that $\Gamma$ is $k$-set-homogeneous if any two isomorphic induced subgraphs of $\Gamma$ of order at most $k$ are equivalent under $\operatorname{Aut}(\Gamma)$, and say that $\Gamma$ is $k$-homogeneous if every isomorphism between any two isomorphic induced subgraphs of order at most $k$ extends to an automorphism of $\Gamma$. Here $\overline{\Gamma}$ denotes the complement of $\Gamma$, $\mathrm{C}_5$ the $5$-cycle, $\mathrm{K}_n$ the complete graph on $n$ vertices, $n\mathrm{K}_m$ the disjoint union of $n$ copies of $\mathrm{K}_m$, $\mathrm{K}_{n[m]}$ the complete multipartite graph with $n$ parts of size $m$, and $\mathrm{K}_n\square \mathrm{K}_n$ the Cartesian product of two copies of $\mathrm{K}_n$. Which statement is equivalent to $\Gamma$ being $k$-set-homogeneous?

$\Gamma$ 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 $\mathrm{C}_5$, $\mathrm{K}_3\square \mathrm{K}_3$, $n\mathrm{K}_m$, or $\mathrm{K}_{n[m]}$; furthermore, $\Gamma$ is homogeneous. (2) $\Gamma$ is the Schla\"{f}li graph on $27$ vertices or its complement; furthermore, $\Gamma$ is $4$-homogeneous but not $5$-homogeneous. (3) $\Gamma$ is either $\mathrm{K}_n\square \mathrm{K}_n$ with $n\ge 4$, or one of the graphs $\mathrm{VO}^{\epsilon}_{2m}(2)$ with $m\ge 1$ and $\epsilon\in\{+,-\}$, $\Gamma(\mathrm{O}_6^-(q))$ with $q$ a prime power, an orbital graph of the Higman--Sims group acting on $100$ points, or an orbital graph of the McLaughlin group acting on $275$ points, or the complement of any such graph; furthermore, $\Gamma$ is $3$-homogeneous but not $4$-homogeneous. (4) $\Gamma$ is an orbital graph of a primitive permutation group of rank $3$ not covered by (1)--(3); furthermore, $\Gamma$ is $2$-homogeneous but not $3$-homogeneous.

Choices:
B. $\Gamma$ is $k$-homogeneous. Moreover, if $\Gamma$ is $k$-homogeneous with $k\ge 2$, then one of the following holds: (1) $k\ge 4$, and $\Gamma$ is one of $\mathrm{C}_5$, $\mathrm{K}_3\square \mathrm{K}_3$, $n\mathrm{K}_m$, or $\mathrm{K}_{n[m]}$; furthermore, $\Gamma$ is homogeneous. (2) $\Gamma$ is the Schla\"{f}li graph on $27$ vertices or its complement; furthermore, $\Gamma$ is $4$-homogeneous but not $5$-homogeneous. (3) $\Gamma$ is either $\mathrm{K}_n\square \mathrm{K}_n$ with $n\ge 4$, or one of the graphs $\mathrm{VO}^{\epsilon}_{2m}(2)$ with $m\ge 1$ and $\epsilon\in\{+,-\}$, $\Gamma(\mathrm{O}_6^-(q))$ with $q$ a prime power, an orbital graph of the Higman--Sims group acting on $100$ points, or an orbital graph of the McLaughlin group acting on $275$ points, or the complement of any such graph; furthermore, $\Gamma$ is $3$-homogeneous but not $4$-homogeneous. (4) $\Gamma$ is an orbital graph of a primitive permutation group of rank $3$ not covered by (1)--(3); furthermore, $\Gamma$ is $2$-homogeneous but not $3$-homogeneous.
C. $\Gamma$ is $k$-homogeneous.
D. $\Gamma$ 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 $\mathrm{C}_5$, $\mathrm{K}_3\square \mathrm{K}_3$, $n\mathrm{K}_m$, or $\mathrm{K}_{n[m]}$; furthermore, $\Gamma$ is homogeneous. (2) $\Gamma$ is the Schla\"{f}li graph on $27$ vertices or its complement; furthermore, $\Gamma$ is homogeneous. (3) $\Gamma$ is either $\mathrm{K}_n\square \mathrm{K}_n$ with $n\ge 4$, or one of the graphs $\mathrm{VO}^{\epsilon}_{2m}(2)$ with $m\ge 1$ and $\epsilon\in\{+,-\}$, $\Gamma(\mathrm{O}_6^-(q))$ with $q$ a prime power, an orbital graph of the Higman--Sims group acting on $100$ points, or an orbital graph of the McLaughlin group acting on $275$ points, or the complement of any such graph; furthermore, $\Gamma$ is $3$-homogeneous but not $4$-homogeneous. (4) $\Gamma$ is an orbital graph of a primitive permutation group of rank $3$ not covered by (1)--(3); furthermore, $\Gamma$ is $2$-homogeneous but not $3$-homogeneous.
E. $\Gamma$ 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 $\mathrm{C}_5$, $\mathrm{K}_3\square \mathrm{K}_3$, $n\mathrm{K}_m$, or $\mathrm{K}_{n[m]}$; furthermore, $\Gamma$ is homogeneous. (2) $\Gamma$ is the Schla\"{f}li graph on $27$ vertices or its complement; furthermore, $\Gamma$ is $4$-homogeneous but not $5$-homogeneous. (3) $\Gamma$ is either $\mathrm{K}_n\square \mathrm{K}_n$ with $n\ge 4$, or one of the graphs $\mathrm{VO}^{\epsilon}_{2m}(2)$ with $m\ge 1$ and $\epsilon\in\{+,-\}$, $\Gamma(\mathrm{O}_6^-(q))$ with $q$ a prime power, an orbital graph of the Higman--Sims group acting on $100$ points, or an orbital graph of the McLaughlin group acting on $275$ points; furthermore, $\Gamma$ is $3$-homogeneous but not $4$-homogeneous. (4) $\Gamma$ is an orbital graph of a primitive permutation group of rank $3$ not covered by (1)--(3); furthermore, $\Gamma$ is $2$-homogeneous but not $3$-homogeneous.


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 whose canonical bundle $K_X$ is semiample, and let $\omega(t)$ be the solution for all $t\ge 0$ of the normalized K\"ahler-Ricci flow
\[
\frac{\partial}{\partial t}\omega(t)=-\operatorname{Ric}(\omega(t))-\omega(t),\qquad \omega(0)=\omega_0.
\]
Let $Y$ be the canonical model associated to $K_X$, let $Y^\circ$ denote the smooth locus on which the canonical metric $\omega_{\rm can}$ is defined, and let $(Y,d_{\rm can})$ be the metric completion of the length metric induced by $\omega_{\rm can}$ on $Y^\circ$. As $t\to\infty$, which limiting statement holds for $(X,\omega(t))$ in the Gromov-Hausdorff topology?

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 $(Y^\circ,\omega_{\rm can})$, so the conjectured Gromov-Hausdorff limit is realized.

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; in particular, the limit is the smooth locus endowed with the canonical K\"ahler metric.
C. As $t\to\infty$, the metric spaces $(X,\omega(t))$ admit subsequences that converge in the Gromov-Hausdorff topology to $(Y,d_{\rm can})$.
D. As $t\to\infty$, the metric spaces $(X,\omega(t))$ converge in the Gromov-Hausdorff topology to a compact metric space canonically homeomorphic to $Y$, but the limit need not coincide with the metric completion $(Y,d_{\rm can})$ 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 $Y$ after identifying $X$ with $Y$ through the canonical map.