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Mengyu Cao

Publications and source records attributed to Mengyu Cao.

At least 19 recordsLinked to original sources
arXiv2026

A Near-Optimal Linear Range for the Erdős Matching Conjecture

The Erdős Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\ge2$, there is $s_0(k)$ such that, whenever $s\ge s_0(k)$ and $n\ge(k+1)s$, every $\mathcal{F}\subseteq\binom{[n]}k$ with $ν(\mathcal{F})\le s$ satisfies \[ |\mathcal{F}|\le\binom nk-\binom{n-s}k, \] with equality only for the family of all $k$-sets meeting a fixed $s$-set. This improves the best previous general linear coefficient from $(5k-2)/3$ to $k+1$. In particular, the parameterized form of our argument further lowers the coefficient to $k+0.6$ for $k\ge5$. Since the two conjectured constructions exchange asymptotic dominance at $n=(ρ_k+o(1))s$ for a coefficient $ρ_k\in(k,k+1)$, our range lies less than one unit above the unavoidable barrier. We also prove a stability theorem showing that cover families are the only near-extremal configurations throughout this range. A key ingredient in our proof is a probabilistic rigidity statement which forces near-extremal fractional covers to be almost integral.

math.CO↗
arXiv2026

Extremal Families for Matchings in Permutations

Two permutations $σ,τ\in S_n$ are called disjoint if the composition $στ^{-1}$ has no fixed point. If a family $\mathcal F\subseteq S_n$ contains no $s$ pairwise disjoint permutations, then a simple averaging argument gives $|\mathcal F|\leq(s-1)(n-1)!$. Inozemtsev, Kolupaev and Kupavskii characterized the equality cases in the range $s\leq n/(2^{17}\log n).$ We characterize all equality cases throughout the range $2\le s\le n$: equality holds if and only if $\mathcal F$ is a union of $(s-1)$ pairwise disjoint $1$-cosets. We also prove the linear statement underlying this classification: a real-valued function on $S_n$ has constant sum on every one-factorization if and only if it lies in the span of the indicators of the $1$-cosets. The proof is combinatorial and applies to every order, with a few small orders handled separately.

math.CO↗
arXiv2026

A Sharp Spectral Erdős--Ko--Rado Theorem for Uniform Hypergraphs

The spectral Erdős--Ko--Rado problem asks for the largest adjacency-tensor spectral radius of a $t$-intersecting $k$-uniform family. Keevash, Lenz and Mubayi proved that, for fixed $k,t$ and sufficiently large $n$, the unique extremal family is a full $t$-star, and asked whether such a theorem extends to all $n$. Let $\mathcal{A}_r=\{F\in\binom{[n]}k:|F\cap[t+2r]|\ge t+r\}$ be the Frankl families and write $ρ_r$ for their spectral radii. For $2\le t 2k-t$, we prove that $\mathcal{A}_0$ is spectrally extremal if and only if $ρ_0\geρ_1$; it is unique up to permutation when the inequality is strict, whereas $\mathcal{A}_0$ and $\mathcal{A}_1$ are both extremal at equality. The layerwise pull used in the Ahlswede--Khachatrian cardinality proof is not applicable here: applied directly, it may decrease the spectral radius. Our proof instead pulls all boundary layers simultaneously and applies Perron tail symmetrization. It follows that $\mathcal{A}_0$ is uniquely extremal for $n\ge (t+1)(k-t+1)+\lceil(t+1)\log(t+1)\rceil$; the leading coefficient $t+1$ is best possible for fixed $t$. We also determine all extremal structures for $t=1$ throughout the range $n\ge2k$.

math.CO↗
arXiv2026

On Erdős--Ko--Rado and Hilton--Milner Theorems for Direct Products

We investigate $t$-intersecting families in direct-product set systems obtained by prescribing the number of selected elements in each part of a partitioned ground set. For a finite union of layers, we prove an Erdős--Ko--Rado result under coordinatewise linear part-size conditions. As an application, we establish a new range of parameters for which a conjecture of Frankl et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{155} (2018), 493--502] holds. Under an explicit polynomial large-part hypothesis, we also characterize the maximum nontrivial $t$-intersecting families for the single-layer setting. In particular, for $t=1$, this answers the problem of Kwan et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{156} (2018), 44--60] asking for a classification of all extremal families, including the possible non-shifted maximizers.

math.CO↗
arXiv2026

Entropy Transference for Rainbow-$H$-Free Colourings of Random Graphs

Let $H$ be a fixed graph with $e(H)\ge3$ that contains two adjacent edges, and let $\ell\ge e(H)$ be fixed. We establish an entropy-transference principle for rainbow-$H$-free edge-colourings of the binomial random graph at the natural scale $p=n^{-1/m_2(H)}$. Writing $R_{H,\ell}(G)$ for the number of such colourings and $λ(H,\ell)$ for the rainbow entropy--Turán density on complete graphs, we show that, with high probability, the per-edge logarithmic counting rate can be made arbitrarily close to $\log\ell$ below a sufficiently small constant multiple of this scale, and arbitrarily close to $λ(H,\ell)$ above a sufficiently large constant multiple. Thus the dense-side counting rate on a sparse random host is governed exactly by a deterministic entropy--Turán parameter on complete graphs. We further investigate this parameter, obtaining partial exact evaluations, corresponding counting-stability results, and its first-order asymptotic behaviour as the number of colours tends to infinity. This extends the random Gallai-colouring transition from triangles to every fixed non-matching graph containing at least three edges, and provides a general mechanism for transferring complete-graph template entropy to sparse random hosts.

math.CO↗
arXiv2026

Counting sunflowers with restricted matching number

For $\mathcal H\subseteq\binom{[n]}k$, let $ν(\mathcal H)$ denote its matching number and let $d_{\mathcal H}(E)$ denote the codegree of a $(k-1)$-set $E$. We study the codegree moments $ co_p(\mathcal H)=\sum_{E\in\binom{[n]}{k-1}}d_{\mathcal H}(E)^p $ and the number of copies of the $k$-uniform $l$-petal sunflower $S_{k,l}^{k-1}$ whose core has size $k-1$. For fixed $k,s,l$ and $p$, and all sufficiently large $n$, we prove that among all $k$-uniform families with matching number at most $s$, both quantities are uniquely maximized by \[ \mathcal H_{n,k,s}=\left\{F\in\binom{[n]}k:F\cap[s]\neq\emptyset\right\}. \] Thus the sunflower result is the generalized Turán problem $\operatorname{ex}_k(n,S_{k,l}^{k-1},M_{s+1})$. When $k=3$, we make the threshold effective: for every integer $p\geq1$, the codegree-moment conclusion holds for $n\geq6s+3$. The proof combines a Hilton--Milner-type stability theorem of Frankl and Kupavskii with a high-codegree-shadow reduction and a convex moment estimate.

math.CO↗
arXiv2026

Erdős--Ko--Rado and Hilton--Milner Theorems in the Partition Lattice

Let $M_n=M(K_{n+1})$ be the graphic matroid of the complete graph, and let $\mathcal{F}_k(M_n)$ be its rank-$k$ flats. We study families $\mathcal{A}\subseteq\mathcal{F}_k(M_n)$ satisfying $\mathrm{rk}(A\wedge B)\ge t$ for all $A,B\in\mathcal{A}$. For $t=1$, this problem is exactly equivalent to Czabarka's partition-EKR conjecture, first introduced in print by P.~L. Erdős and L.~A. Székely~\cite{ErdosSzekelyHigher}. We prove the corresponding Erdős--Ko--Rado theorem in the explicit linear range $n+1\ge8k$, giving a constant-factor advance toward the conjectured sharp range $n\ge2k$. For every fixed $t$, we further prove an Erdős--Ko--Rado theorem under an explicit condition of order $O_t(k^2)$ on the block number $n+1-k$, with equality only for a full $t$-star. We also determine the largest nontrivial intersecting families under an explicit $O(k^6)$ threshold and characterize the unique extremal family up to isomorphism.

math.CO↗
arXiv2026

Convex Transference for Degree Powers in Extremal Set Systems

For a family $\mathcal{F}\subseteq\binom{[n]}k$ and $R\in\binom{[n]}r$, let $d_{\mathcal{F}}(R)=|\{F\in\mathcal{F}:R\subseteq F\}|$ and $\ell_{r,p}(\mathcal{F})=\sum_{R\in\binom{[n]}r}d_{\mathcal{F}}(R)^p$; at the codegree level, write $co_p(\mathcal{F})=\ell_{k-1,p}(\mathcal{F})$. We introduce a new convex-transference method for degree-power extremal problems and develop it into a reusable input--transfer--rigidity framework independent of any particular set-system problem. We give three exact applications. First, a full $t$-star maximizes $co_p$ among $t$-intersecting families for every real $p\geq2$ in the sharp range $n\geq(t+1)(k-t+1)$, with all equality cases determined. This extends the Wu--Zhang quadratic theorem to real exponents and answers a problem of Zhou--Yuan throughout the sharp Erdős--Ko--Rado range. Second, if $n\geq2k$, a full point-star maximizes $\ell_{r,p}$ for every $1\leq r\leq k-1$ and real $p\geq2$, again with complete equality classification; thus the framework is not confined to codegrees. Third, if $ν(\mathcal{F})\leq s$ and $n\geq(2s+1)k-s$, then for every real $p\geq1$, $co_p(\mathcal{F})$ is uniquely maximized, up to isomorphism, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or nonexplicit sufficiently-large assumptions with an explicit linear range valid for arbitrary uniformity.

math.CO↗
arXiv2026

Projective Ore-Degree Conditions for Intersection Theorems in Vector Spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, and let $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$. The \emph{projective Ore-degree} of $\mathcal F$ is the minimum, over all $k$-subspaces $S\notin\mathcal F$, of the sum of the $\mathcal F$-degrees of the projective points contained in $S$. We prove sharp projective Ore analogues of the vector-space Erdős--Ko--Rado and Hilton--Milner theorems. The Ore--Erdős--Ko--Rado theorem holds for $n\ge2k+1$, with equality only for a full point-star. For nontrivial intersecting families, we determine the sharp Ore--Hilton--Milner threshold, together with the complete equality classification, when $q\ge3$ and $n\ge2k+1$, or when $q\ge2$ and $n\ge2k+2$. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s\ge3$ and $n\ge(2s-1)k-s+4$, and derive a multicolour Ramsey consequence.

math.CO↗
arXiv2026

Matchings and Near-Optimal 2-Factor Packings in Percolated Vertex-Transitive Graphs

Let $G$ be a connected simple vertex-transitive graph on $n$ vertices with degree $d$, and let $G_p$ be the random spanning subgraph obtained by retaining each edge of $G$ independently with probability $p$. Put $q:=1-p$. Motivated by a conjecture of Bedert, Draganić, Müyesser, and Pavez-Signé on Hamilton cycles in percolated Cayley graphs, we establish the corresponding matching and $2$-factor statements uniformly over the larger class of all connected vertex-transitive host graphs. For every $A>0$, if $q^d\le n^{-(5A+250)},$ then, with probability at least $1-n^{-A}$, the graph $G_p$ has a perfect matching when $n$ is even and is factor-critical when $n$ is odd. Separately, if $0<ε<1$ and $ ε^2pd\ge64(A+6)\log(2n), $ then, with probability at least $1-n^{-A}$, the graph $G_p$ contains at least \[ \left\lfloor\frac{(1-ε)pd}{2}\right\rfloor \] pairwise edge-disjoint spanning $2$-factors. Moreover, if $pd/\log n\to\infty$, then \[ ν_2(G_p)=(1+o(1))\frac{pd}{2} \] with high probability, which is asymptotically optimal, where $ν_2(G)$ is the maximum number of pairwise edge-disjoint spanning 2-factors in $G$. Thus logarithmic-order percolation already forces these two factor-theoretic consequences of Hamiltonicity beyond the Cayley setting.

math.CO↗
arXiv2026

Generalizations of the Erdős Matching Conjecture for the $t$-Matching Number

We write finite set systems as uniform hypergraphs. A \emph{$t$-matching} in a $k$-uniform hypergraph is a set of hyperedges any two of which intersect in fewer than $t$ vertices. The maximum size of such a set is the \emph{$t$-matching number} and is denoted by $ν_t$. We study the maximum number of hyperedges in a $k$-uniform hypergraph on $[n]$ with prescribed $t$-matching number. This gives a hypergraph analogue of the Erdős Matching Conjecture. We also determine the second largest maximal structure with $ν_t(\mathcal{F})=s$, extending work of Frankl and Kupavskii \cite{frankl2016two}. And, we obtain the extremal $G$-free induced subgraphs of generalized Kneser graph, generalizing Alishahi's results in \cite{alishahi2018extremal}.

math.CO↗
arXiv2026

On degree bounds of $k$-uniform hypergraphs with bounded matching number

We study the connection between the degree sequence of a $k$-uniform hypergraph and the size of its largest matching. Let $\mathcal{F}$ be a $k$-uniform hypergraph on $n$ vertices and let $d_1(\mathcal{F})\ge d_2(\mathcal{F})\ge\cdots\ge d_n(\mathcal{F})$ be the vertex degrees arranged in non-increasing order. For integers $k\ge 2$, $s\ge 2$ and $n > 2sk$, we prove that if the $2sk$-th largest degree satisfies $d_{2sk} > \binom{n-1}{k-1} - \binom{n-s}{k-1},$ then $\mathcal{F}$ contains a matching of size at least $s$. Moreover, by relaxing the range of $n$, we obtain the same bound for the $(k+2s-2)$-th largest degree vertex. Note that the number $k+2s-2$ is optimal. For a $k$-set of vertices $S \subseteq [n]$, the degree of $S$ is defined as $°(S) = \sum_{v \in S} °(v)$, and the minimum of $°(S)$ over all non-edge $k$-subsets $S \notin E(\mathcal{F})$ of $V(\mathcal{F})$ is the \textit{Ore-degree} of $\mathcal{F}$, denoted by $σ_k(\mathcal{F})$. Balogh, Palmer and Raeisi proved: for $s \ge 2$ and $n \ge 3k^2(s-1)$, if $σ_k(\mathcal{F}) > k\left(\binom{n-1}{k-1} - \binom{n-s}{k-1}\right),$ then $\mathcal{F}$ contains a matching of size $s$. They also conjectured that the result holds when $n > sk$. As a corollary, we prove that the bound on $n$ can be taken to be linear in $sk$ ($ n \geq 3sk $).

math.CO↗
arXiv2026

MapSatisfyBench: Benchmarking Satisfaction-Aware Map Agents through Behavior-Grounded Implicit Decision Factors

Large language model agents are increasingly integrated into map services. Since map services are embedded in everyday-life scenarios rather than professional task settings, users often express their needs informally, resulting in underspecified queries with many unspoken needs, namely, implicit decision factors that are critical for user satisfaction. Although clarification is an effective way to mitigate this issue, it increases user burden in daily interaction, and a capable agent should first proactively recover such factors from available information sources. However, evaluating this ability is challenging. The first challenge is to determine which implicit decision factors are suitable for evaluation. A factor is evaluable only if it affects user acceptance and can be recovered from information available to the agent before it responds. Second, user satisfaction cannot be reliably represented by a single reference answer, requiring a benchmark that converts satisfaction-relevant factors into objective and quantifiable evaluation targets. To address these challenges, we propose a restore-identify-filter framework that reconstructs complete user needs from behavior-chain evidence, identifies implicit decision factors, and retains only those supported by pre-query evidence. Building on this methodology, we construct MapSatisfyBench from large-scale, real-world anonymized user data and annotate ground truth from five dimensions and enables full-chain evaluation of satisfaction-aware map agents. Experiments show that current agents generally perform well on explicit task completion, but remain limited in satisfying implicit decision factors and proactively acquiring the evidence needed for satisfaction-aware decisions. These findings establish MapSatisfyBench as a benchmark for shifting map-agent evaluation from task completion toward satisfaction-aware spatial decision making.

cs.AI↗
arXiv2025

Cross-intersection theorems for uniform partitions of finite sets

A set partition is $c$-uniform if every block has size $c$. Two families of $c$-uniform partitions of a finite set are said to be cross $t$-intersecting if two partitions from different families share at least $t$ blocks. In this paper, we establish some product-type extremal results for such cross $t$-intersecting families. Our results yield an Erdős-Ko-Rado theorem and a Hilton-Milner theorem for uniform set partitions. Additionally, cross $t$-intersecting families with the maximum sum of their sizes are also characterized.

math.CO↗
arXiv2025

Treewidth of generalized Hamming graph, bipartite Kneser graph and generalized Petersen graph

Let $t,q$ and $n$ be positive integers. Write $[q] = \{1,2,\ldots,q\}$. The generalized Hamming graph $H(t,q,n)$ is the graph whose vertex set is the cartesian product of $n$ copies of $[q]$ ($q\ge 2$), where two vertices are adjacent if their Hamming distance is at most $t$. In particular, $H(1,q,n)$ is the well-known Hamming graph and $H(1,2,n)$ is the hypercube. In 2006, Chandran and Kavitha described the asymptotic value of $tw(H(1,q,n))$, where $tw(G)$ denotes the treewidth of $G$. In this paper, we give the exact pathwidth of $H(t,2,n)$ and show that $tw(H(t,q,n)) = Θ(tq^n/\sqrt{n})$ when $n$ goes to infinity. Based on those results, we show that the treewidth of the bipartite Kneser graph $BK(n,k)$ is $\binom{n}{k} - 1$ when $n$ is sufficiently large relative to $k$ and the bounds of $tw(BK(2k+1,k))$ are given. Moreover, we present the bounds of the treewidth of the generalized Petersen graph.

math.CO↗
arXiv2025

Edge pancyclic Cayley graphs on symmetric group

We study the derangement graph $Γ_n$ whose vertex set consists of all permutations of $\{1,\ldots,n\}$, where two vertices are adjacent if and only if their corresponding permutations differ at every position. It is well-known that $Γ_n$ is a Cayley graph, Hamiltonian and Hamilton-connected. In this paper, we prove that for $n \geq 4$, the derangement graph $Γ_n$ is edge pancyclic. Moreover, we extend this result to two broader classes of Cayley graphs defined on symmetric group.

math.CO↗
arXiv2025

Simplices in $t$-intersecting families for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq {V\brack k}$ is called $k$-uniform $r$-wise $t$-intersecting if for any $F_1, F_2, \dots, F_r \in \mathcal{F}$, we have $\dim\left(\bigcap_{i=1}^r F_i \right) \geq t$. An $r$-wise $t$-intersecting family $\{X_1, X_2, \dots, X_{r+1}\}$ is called a $(r+1,t)$-simplex if $\dim\left(\bigcap_{i=1}^{r+1} X_i \right) < t$, denoted by $Δ_{r+1,t}$. Notice that it is usually called triangle when $r=2$ and $t=1$. For $k \geq t \geq 1$, $r \geq 2$ and $n \geq 3kr^2 + 3krt$, we prove that the maximal number of $Δ_{r+1,t}$ in a $k$-uniform $r$-wise $t$-intersecting subspace family of $V$ is at most $n_{t+r,k}$, and we describe all the extreme families. Furthermore, we have the extremal structure of $k$-uniform intersecting families maximizing the number of triangles for $n\geq 2k+9$ as a corollary.

math.CO↗
arXiv2024

Hilton-Milner theorem for $k$-multisets

Let $ k, n \in \mathbb{N}^+ $ and $ m \in \mathbb{N}^+ \cup \{\infty \} $. A $ k $-multiset in $ [n]_m $ is a $ k $-set whose elements are integers from $ \{1, 2, \ldots, n\} $, and each element is allowed to have at most $ m $ repetitions. A family of $ k $-multisets in $ [n]_m $ is said to be intersecting if every pair of $ k $-multisets from the family have non-empty intersection. In this paper, we give the size and structure of the largest non-trivial intersecting family of $ k $-multisets in $ [n]_m $ for $ n \geq k + \lceil k/m \rceil $. In the special case when $m=\infty$, our result gives rise to an unbounded multiset version for Hilton-Milner Theorem given by Meagher and Purdy. Furthermore, our main theorem unites the statements of the Hilton-Milner Theorem for finite sets and unbounded multisets.

math.CO↗