arXiv subjects

Haixiang Zhang

Publications and source records attributed to Haixiang Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Extremal Families for Matchings in Permutations

Two permutations $\sigma,\tau\in S_n$ are called disjoint if the composition $\sigma\tau^{-1}$ 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 Erd\H{o}s--Ko--Rado Theorem for Uniform Hypergraphs

The spectral Erd\H{o}s--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 $\rho_r$ for their spectral radii. For $2\le t 2k-t$, we prove that $\mathcal{A}_0$ is spectrally extremal if and only if $\rho_0\ge\rho_1$; 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

Efficient Poisson Subsampling for the Partially Linear Additive Cox Model

To address the computational and storage challenges often encountered in large-scale survival data analysis, we propose an efficient Poisson subsampling method for the partially linear additive Cox model. This model provides a flexible yet interpretable framework by incorporating linear covariate effects, additive nonparametric components for nonlinear covariates, and a nonparametric baseline hazard function. The proposed method adopts B-spline basis functions to approximate the nonparametric components and employs the decorrelated score technique to construct a Poisson subsampling-based estimation equation, based on which we establish the asymptotic normality of the resulting estimator and derive the optimal subsampling probabilities according to the L-optimality criterion. Furthermore, we design a two-step adaptive algorithm for practical implementation. The proposed approach enables computationally efficient statistical inference for large-scale survival analysis without processing the full dataset. We validate the performance of the proposed method through extensive simulation studies and a real-world application to a lymphoma cancer dataset, demonstrating its efficiency and accuracy in large-scale settings.

stat.ME↗

arXiv2026

A Near-Optimal Linear Range for the Erd\H{o}s Matching Conjecture

The Erd\H{o}s 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 $\nu(\mathcal{F})\le s$ 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 $n=(\rho_k+o(1))s$ for a coefficient $\rho_k\in(k,k+1)$, 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

Erd\H{o}s--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. Erd\H{o}s and L.~A. Sz\'ekely~\cite{ErdosSzekelyHigher}. We prove the corresponding Erd\H{o}s--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 Erd\H{o}s--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

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 $\lambda(H,\ell)$ for the rainbow entropy--Tur\'an 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 $\lambda(H,\ell)$ above a sufficiently large constant multiple. Thus the dense-side counting rate on a sparse random host is governed exactly by a deterministic entropy--Tur\'an 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

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 Erd\H{o}s--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 $\nu(\mathcal{F})\leq s$ 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 Erd\H{o}s--Ko--Rado and Hilton--Milner theorems. The Ore--Erd\H{o}s--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

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 $\deg(S) = \sum_{v \in S} \deg(v)$, and the minimum of $\deg(S)$ 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 $\sigma_k(\mathcal{F})$. Balogh, Palmer and Raeisi proved: for $s \ge 2$ and $n \ge 3k^2(s-1)$, if $\sigma_k(\mathcal{F}) > k\left(\binom{n-1}{k-1} - \binom{n-s}{k-1}\right),$ 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

Algorithm for finding vertex-edge domination number on graphs with bounded treewidth and related problems on planar graphs

Given a graph $G=(V,E)$, a vertex $u \in V$ {\em ve-dominates} all edges incident to any vertex of $N_G[u]$. A set $S \subseteq V$ is a {\em ve-dominating set} if for all edges $e\in E$, there exists a vertex $u\in S$ such that $u$ ve-dominates $e$. The minimum cardinality among all ve-dominating sets is known as the \textit{vertex-edge domination number} (or simply ve-domination number) and denoted by $\gamma_{ve}(G)$. Finding a minimum ve-dominating set was proved to be NP-complete. Restricted to trees, the problem admits a linear-time algorithm. Treewidth is a commonly used parameter for solving NP-hard problems. In this paper, we present a polynomial-time algorithm for finding a minimum ve-dominating set on graphs with bounded treewidth. Moreover, we show that the treewidth of a planar graph $G$ with ve-domination number $\gamma_{ve}(G)$ is $O(\sqrt{\gamma_{ve}(G)})$ and present an $O(c^{\sqrt{k}}|V(G)|)$-time algorithm for the $k$-ve-domination problem on planar graphs.

math.CO↗

arXiv2026

Counting sunflowers with restricted matching number

For $\mathcal H\subseteq\binom{[n]}k$, let $\nu(\mathcal H)$ 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 Tur\'an 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↗

arXiv2025

The maximum sum of sizes of non-empty cross $L$-intersecting families

Let $n$, $r$, and $k$ be positive integers such that $k, r \geq 2$, $L$ a non-empty subset of $[k]$, and $\mathcal{F}_i \subseteq \binom{[n]}{k}$ for $1 \leq i \leq r$. We say that non-empty families $\mathcal{F}_1, \mathcal{F}_2, \ldots, \mathcal{F}_r$ are $r$-cross $L$-intersecting if $\left| \bigcap_{i=1}^r F_i \right| \in L$ for every choice of $F_i \in \mathcal{F}_i$ with $1 \leq i \leq r$. They are called pairwise cross $L$-intersecting if $|A \cap B| \in L$ for all $A \in \mathcal{F}_i$, $B \in \mathcal{F}_j$ with $i \neq j$. If $r=2$, we simply say cross $L$-intersecting instead of $2$-cross $L$-intersecting or pairwise cross $L$-intersecting. In this paper, we determine the maximum possible sum of sizes of non-empty cross $L$-intersecting families $\mathcal{F}_1$ and $\mathcal{F}_2$ for all admissible $n$, $k$, and $L$, and we characterize all the extremal structures. We also establish the maximum value of the sum of sizes of families $\mathcal{F}_1, \dots, \mathcal{F}_r$ that are both pairwise cross $L$-intersecting and $r$-cross $L$-intersecting, provided $n$ is sufficiently large and $L$ satisfies certain conditions. Furthermore, we characterize all such families attaining the maximum total size.

math.CO↗

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 Erd\H{o}s-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

Counting induced subgraphs with given intersection sizes

Let $F$ be a graph of order $r$. In this paper, we study the maximum number of induced copies of $F$ with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subseteq[0,r-1]$ be an integer set with $s\not\in\{1,r\}$. Let $\Psi_r(n,F,L)$ be the maximum number of induced copies of $F$ in an $n$-vertex graph, where the induced copies of $F$ are $L$-intersecting as a family of $r$-subsets, i.e., for any two induced copies of $F$, the size of their intersection is in $L$. Helliar and Liu initiated a study of the function $\Psi_r(n,K_r,L)$. Very recently, Zhao and Zhang improved their result and showed that $\Psi_r(n,K_r,L)=\Theta_{r,L}(n^{s})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression. In this paper, we show that $\Psi_r(n,F,L)=o_{r,L}(n^{s})$ when $\ell_1,\dots,\ell_s,r$ do not form an arithmetic progression. We study the asymptotical result of $\Psi_r(n,C_r,L)$, and determined the asymptotically optimal result when $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and take certain values. We also study the generalized Tur\'an problem, determining the maximum number of $H$, where the copies of $H$ are $L$-intersecting as a family of $r$-subsets. The entropy method is used to prove our results.

math.CO↗

arXiv2025

Generalizations of the Erd\H{o}s 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 $\nu_t$. 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 Erd\H{o}s Matching Conjecture. We also determine the second largest maximal structure with $\nu_t(\mathcal{F})=s$, 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↗

arXiv2025

Turán type problems for a fixed graph and a linear forest

Let $\mathscr{F}$ be a family of graphs. A graph $G$ is $\mathscr{F}$-free if $G$ does not contain any $F\in \mathscr{F}$ as a subgraph. The Turán number, denoted by $ex(n, \mathscr{F})$, is the maximum number of edges in an $n$-vertex $\mathscr{F}$-free graph. Let $F $ be a fixed graph with $ χ(F) \geq 3 $. A forest $H$ is called a linear forest if all components of $H$ are paths. In this paper, we determined the exact value of $ex(n, \{H, F\}) $ for a fixed graph $F$ with $χ(F)\geq 3$ and a linear forest $H$ with at least $2$ components and each component with size at least $3$.

math.CO↗

arXiv2025

Exact Recovery Guarantees for Parameterized Nonlinear System Identification Problem under Sparse Disturbances or Semi-Oblivious Attacks

In this work, we study the problem of learning a nonlinear dynamical system by parameterizing its dynamics using basis functions. We assume that disturbances occur at each time step with an arbitrary probability $p$, which models the sparsity level of the disturbance vectors over time. These disturbances are drawn from an arbitrary, unknown probability distribution, which may depend on past disturbances, provided that it satisfies a zero-mean assumption. The primary objective of this paper is to learn the system's dynamics within a finite time and analyze the sample complexity as a function of $p$. To achieve this, we examine a LASSO-type non-smooth estimator, and establish necessary and sufficient conditions for its well-specifiedness and the uniqueness of the global solution to the underlying optimization problem. We then provide exact recovery guarantees for the estimator under two distinct conditions: boundedness and Lipschitz continuity of the basis functions. We show that finite-time exact recovery is achieved with high probability, even when $p$ approaches 1. Unlike prior works, which primarily focus on independent and identically distributed (i.i.d.) disturbances and provide only asymptotic guarantees for system learning, this study presents the first finite-time analysis of nonlinear dynamical systems under a highly general disturbance model. Our framework allows for possible temporal correlations in the disturbances and accommodates semi-oblivious adversarial attacks, significantly broadening the scope of existing theoretical results.

math.OC↗

arXiv2025

Distributionally Robust Joint Chance-Constrained Optimal Power Flow using Relative Entropy

Designing robust algorithms for the optimal power flow (OPF) problem is critical for the control of large-scale power systems under uncertainty. The chance-constrained OPF (CCOPF) problem provides a natural formulation of the trade-off between the operating cost and the constraint satisfaction rate. In this work, we propose a new data-driven algorithm for the CCOPF problem, based on distributionally robust optimization (DRO). \revise{We show that the proposed reformulation of the distributionally robust chance constraints is exact, whereas other approaches in the CCOPF literature rely on conservative approximations. We establish out-of-sample robustness guarantees for the distributionally robust solution and prove that the solution is the most efficient among all approaches enjoying the same guarantees.} We apply the proposed algorithm to the the CCOPF problem and compare the performance of our approach with existing methods using simulations on IEEE benchmark power systems.

math.OC↗