arXiv subjects

Jiasheng Zeng

Publications and source records attributed to Jiasheng Zeng.

At least 19 recordsLinked to original sources

arXiv2026

Turán number of a matching and a Berge triangle

For a fixed graph $G$, an $r$-uniform hypergraph is said to contain a Berge-$G$ if there exists a bijection $f\colon E(G)\to E(\mathcal{H})$ for some subhypergraph $\mathcal{H}$ such that $e\subseteq f(e)$ for every $e\in E(G)$. Motivated by Alon and Frankl's study of Turán problems under bounded matching constraints, we investigate the maximum number of edges in $r$-uniform Berge-$K_3$-free hypergraphs with matching number at most~$s$. We determine the exact Turán numbers for the cases $r=3$ and $r=4$. For $r=3$ and $n \geq 3 s$, we prove that every $n$-vertex Berge- $K_3$-free 3-graph with matching number $s$ has at most $s(n-2 s)$ edges, and we characterize the unique extremal hypergraph attaining equality. For $r=4$ and $n \geq 4 s$, the maximum number of edges is $s\lfloor(n-2 s) / 2\rfloor$, except for the exceptional case $s=1$ and $n \equiv 1(\bmod 4)$, in which the bound is $(n-1) / 2$. As a corollary, our results recover the classical theorem of Győri on Berge-$K_3$-free hypergraphs.

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arXiv2026

Polynomial positivity cones for Coxeter roots and walks in trees

For a finite simple graph $G$ and an integer $k\ge0$, let $w_k(G)$ denote the number of walks of length $k$. We prove the conjecture of Täubig, Weihmann, Kosub, Hemmecke, and Mayr for every finite tree and determine all equality cases. If $T$ has $n\ge1$ vertices, then $n w_{k+1}(T)-2(n-1)w_k(T)\ge0$ for every $k\ge1$; for $n\ge3$, equality holds if and only if $T$ is a star and $k$ is even, whereas for $n=1$ or $n=2$, equality holds for every $k\ge1$. For non-Dynkin trees and even indices, the proof is based on a polynomial positivity cone associated with the adjacency operator of a finite graph and a positive real root of its simply-laced Coxeter system. For finite connected bipartite non-Dynkin graphs, we establish sufficient positivity conditions in terms of Coxeter orbits and inversion sets, and verify these conditions for indicator roots supported on connected induced subtrees. For non-Dynkin trees, this yields the rooted even-index inequality and, after summation, the corresponding global inequality. We also prove that if $G$ is a finite connected bipartite non-Dynkin simple graph, $\varnothing\ne U\subseteq V(G)$, and the subgraph of $G$ induced by $U$ is a tree, then $|U|w_{k+1}(G,U)-2(|U|-1)w_k(G,U)\ge0$ for every $k\ge0$, where $w_k(G,U)$ counts the length-$k$ walks in $G$ whose initial and terminal vertices lie in $U$; the intermediate vertices are unrestricted. The remaining even-index cases for finite Dynkin trees are handled by generating-function recurrences, while the odd-index cases follow from a spectral covariance identity.

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arXiv2026

Compactness of abundance in asymmetric hypergraph removal lemmas

Fix an integer $r\ge 2$ and a finite simple $r$-uniform hypergraph $F$ with at least one edge and no isolated vertices. An $n$-vertex $r$-graph is $ε$-far from being $F$-free if at least $εn^r$ edges must be deleted to destroy every copy of $F$. A finite $r$-graph $H$ is $F$-abundant if there are constants $c,C>0$ such that every sufficiently large $ε$-far host contains at least $cε^C n^{v(H)}$ labelled copies of $H$. A family is $F$-abundant when one member has this lower bound in each host, although the member may depend on the host and on $ε$, while $c$ and $C$ are common to the family. We prove that every abundant family contains an abundant member. We prove the analogous coloured theorem for $F$-partite hosts containing edge-disjoint part-respecting copies of $F$ such that every vertex lies in at least $εn^{r-1}$ of them. The case $r=2$ yields the coloured and uncoloured graph compactness theorems, answers Question 5.2 of Girão, Hurley, Illingworth and Michel, and proves their Conjecture 5.1 [J. Lond. Math. Soc., 2024]. We also obtain an explicit bound $\lfloor 2r(C+1)\rfloor$ for the order of the non-isolated core of a selected witness. Moreover, we give several applications. For example, we construct translation-invariant linear systems from abundant coloured hypergraphs, obtain a square-root bound for an equation associated with a cycle of bounded length, give a one-sided tester based on one fixed graph when distance from the property gives a polynomial lower bound on distance from being $F$-free, and prove that no algorithm decides whether a family of finite simple graphs enumerated by a Turing machine is $K_3$-abundant.

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arXiv2026

Tight Bound for Nikiforov's Spectral Even-Cycle Conjecture

Nikiforov conjectured that, for every fixed $k\ge2$ and all sufficiently large $n$, the unique $n$-vertex $C_{2k+2}$-free graph with maximum adjacency spectral radius is $S^+_{n,k}$, where $S_{n,k}=K_k\vee\overline K_{n-k}$ and $S^+_{n,k}$ is obtained from $S_{n,k}$ by adding one edge inside the independent part. Cioabă, Desai and Tait proved this conjecture for $n\ge k^{O(k)}$. Later, Li and Ning raised the problem of determining the optimal exponent $γ=γ(k)$ such that the same conclusion holds for $n\ge Ω(k^{γ(k)})$. We prove a stronger uniform theorem for Nikiforov's matrices $A_α(G)=αD(G)+(1-α)A(G)$. More precisely, for every $ε>0$ there are constants $C_ε$ and $k_ε$ such that for all $0\leα\le1-ε$, $k\ge k_ε$ and $n\ge C_εk$, every $n$-vertex $C_{2k+2}$-free graph $G$ satisfies $ρ_α(G)\leρ_α(S^+_{n,k})$, with equality if and only if $G\cong S^+_{n,k}$. In particular, the case $α=0$ answers the problem of Li and Ning, and the $A_α$-spectral even-cycle threshold is linear in $k$, uniformly for all $α$ bounded away from $1$. Our proof introduces a weighted rooted Erdős--Gallai type path lemma, which may be of independent interest in Perron-vector methods for spectral extremal graph problems. The same method also yields asymptotically tight $A_α$-spectral bounds for two local forbidden-subgraph families, namely $(K_1\vee P_\ell)$-free graphs and $F_s$-free graphs, where $F_s$ denotes the friendship graph.

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arXiv2026

Asymptotic Uniformity of Permanents of Random Matrices over Finite Fields of Odd Characteristic

Let $q$ be an odd prime power, and let $A_n=(a_{ij})\in\mathbb F_q^{n\times n}$ be a random matrix whose entries are independent and uniformly distributed on $\mathbb F_q$. The permanent of $A_n$ is defined by $\operatorname{per}(A_n)=\sum_{σ\in S_n}\prod_{i=1}^n a_{i,σ(i)}$, where $S_n$ denotes the symmetric group on $[n]$. Ghasemi, Gross, and Kopparty conjectured the zero-mass asymptotic $\Pr[\operatorname{per}(A_n)=0]=1/q+o(1)$ for every fixed odd prime power $q$, and Hunter, Kwan, and Sauermann subsequently stated its equivalent full-distribution formulation: for every fixed $q$ and every $x\in\mathbb F_q$, \[ \lim_{n\to\infty}\Pr[\operatorname{per}(A_n)=x]=\frac1q. \] In this paper, we prove this conjecture. More precisely, we prove that there is an absolute constant $C>0$ such that \[\frac12\sum_{x\in\mathbb F_q}\left|\Pr[\operatorname{per}(A_n)=x]-\frac1q\right|\le C\frac{\log n}{n}\] for every odd prime power $q$ and every $n\ge 7$. The estimate is uniform in $q$, so the conclusion remains valid for every sequence $q=q(n)$ of odd prime powers.

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arXiv2026

Every fork-free graph is perfectly weight divisible

A graph $G$ is \emph{perfectly weight divisible} if, for every positive integral weight function on $V(G)$ and every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. Perfect divisibility and its weighted form provide a natural approach to polynomial $χ$-boundedness. A \emph{fork}, also known as a \emph{chair}, is the graph obtained from a claw by subdividing one of its edges once. In this paper, we prove that every fork-free graph is perfectly weight divisible. As a consequence, we confirm a conjecture of Sivaraman that every fork-free graph is perfectly divisible.

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arXiv2026

A Single-Exponential Erdős--Hajnal Bound for Graphs of Bounded VC-Dimension

A homogeneous set in a graph is a clique or a stable set. The Erdős--Hajnal conjecture states that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a homogeneous set of size at least $n^c$. Nguyen, Scott and Seymour proved that for every $d>0$, graphs of VC-dimension at most $d$ have the Erdős--Hajnal property, confirming a conjecture of Fox, Pach and Suk. In particular, they showed that every such $n$-vertex graph contains a homogeneous set of size at least $n^{η_d}$ for some $η_d\ge 2^{-2^{O(d)}}$. In this paper, we give a sharper quantitative bound on the homogeneous sets in graphs of VC-dimension at most $d$, showing that one may take $ η_d\ge (Cd)^{-d}, $ where $C$ is an absolute constant. Equivalently, every graph $G$ of VC-dimension at most $d$ satisfies \[ \max\{ω(G),α(G)\}\ge |G|^{(Cd)^{-d}}. \] Our proof refines the iterative sparsification method of Nguyen, Scott and Seymour. The main enhancement is to apply the VC-dimension assumption directly, which gives a more efficient induction and thus improves the dependence on $d$. We also derive quantitative consequences for polynomial Rödl subgraphs, hypergraph Ramsey bounds under bounded VC-dimension, induced-free and viral formulations, tournaments, NIP and semi-algebraic graphs, Boolean combinations of relations of bounded VC-dimension, graphs whose adjacency matrices have bounded rank, graphs of bounded sign-rank, and graphs defined by dot-product threshold representations.

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arXiv2026

Paths of Odd Order in Graphs with Given Edge Density

We determine the asymptotic maximum number of unlabelled copies of $P_{2r+1}$ in graphs with prescribed edge density, where $r\ge1$ is fixed and $P_{2r+1}$ denotes the path on $2r+1$ vertices. If an $n$ vertex graph $G$ has edge density $c=2e(G)/n^2$, then the maximum is $\frac12S_r(c)n^{2r+1}+O(n^{2r})$ for $0<c\le c_r$, and $\frac12c^{r+1/2}n^{2r+1}+O(n^{2r})$ for $c_r\le c<1$, where $S_r(c)$ is the value given by the quasi-star construction and $c_r\in(0,1)$ is an explicit algebraic transition point. Thus the quasi-star construction is asymptotically extremal below the transition, while the quasi-clique construction is asymptotically extremal above the transition. This extends the quasi-star versus quasi-clique theorem of Ahlswede and Katona for $P_3$ and the theorem of Nagy for $P_5$ to all paths with an odd number of vertices. The proof reduces the problem to threshold graphons and then to two endpoint families. The three-step endpoint is handled by reducing the required inequality to coefficient nonnegativity in a Bernstein expansion, which is proved by a direct combinatorial argument.

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arXiv2026

Finite-Kernel Extremizers in Sparse Extremal Graph Counting

We develop a finite-kernel framework for sparse extremal graph counting. The problems considered here ask for the maximum number of copies or homomorphisms of a fixed graph under sparse edge constraints. In this regime, the leading term need not be governed by a single dense block. Instead, the extremal mass may be supported on several interacting asymptotic scales. Our framework identifies these scales via a finite-dimensional linear program, separates the leading contributions through a finite state decomposition, and synchronizes or realizes them inside a finite kernel. We apply this framework in three settings. First, we prove the sparse threshold conjecture of Day and Sarkar for graphons. For every fixed graph $H$ without isolated vertices, we prove that \[ \sup_{t(K_2,W)\le β} t(H,W)=β^{|V(H)|-α^*(H)}(C_T(H)+o(1)) \] as $β\to0$, where $α^*(H)$ is the fractional independence number of $H$ and $C_T(H)$ is an explicit sharp constant attained by a three-step threshold graphon. Second, we affirmatively answer a question of Blekherman and Patel by showing that, for every graph $H$, whenever $m\to\infty$ and $m=o(n^{3/2})$, threshold graphs asymptotically maximize $\hom(H,G)$ among all graphs with at most $n$ vertices and at most $m$ edges. Third, Gerbner, Nagy, Patkós, and Vizer conjectured that, among all bipartite graphs with $n$ vertices and $m$ edges, the quasi-complete bipartite graph asymptotically maximizes the number of copies of every fixed bipartite graph $H$ whenever $m=ω(n)$ and $m\le n^2/4$. We disprove this conjecture in the subquadratic range and give the correct order of magnitude in terms of $κ_H(n,m)$, a finite-kernel scale defined by a finite-dimensional variational problem.

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arXiv2026

Proofs of Two Conjectures of Alon on Subgraph Counts

All graphs considered are finite with no isolated vertices. Let $N(m,H)$ be the maximum number of subgraphs of a graph $G$ isomorphic to $H$, taken over all graphs $G$ with $m$ edges. Alon proved that $N(m,H)=Θ_H(m^{γ(H)})$, where $γ(H)=(|V(H)|+D(H))/2$ and $D(H)=\max_{S\subseteq V(H)}(|S|-|N_H(S)|)$, and conjectured [Conjecture 1, Isr. J. Math., 1986] that limit of $N(m,H)/m^{γ(H)}$ exists as $m\to\infty$. We prove this conjecture and identify the limit as $λ(H)=Λ(H)/|\operatorname{Aut}(H)|$, where $Λ(H)$ is characterized by a variational problem over finite cores. We also resolve another conjecture of Alon [Conjecture 2, Isr. J. Math., 1986], which stated that if $H$ is a disjoint union of stars, then for every $m$ an extremal graph attaining $N(m,H)$ may be chosen to be a disjoint union of stars.

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arXiv2026

Sharp Bounds for Guiduli-Type Hereditary Spectral Problems

Guiduli asked in 1996 the following problem concerning the maximum spectral radius of a graph under hereditary density constraints. If an $n$-vertex graph $G$ satisfies $e(H)\le c|V(H)|^2$ for every subgraph $H$ of $G$, must one have $λ(G)\le 2cn$? More generally, what remains true when the exponent $2$ is replaced by a constant less than $2$? We study the natural power-law version of this question for all $1<p\le2$. For $1<p\le 2$, define \[ d_p(G)=\max_{\varnothing\ne S\subseteq V(G)}\frac{e(G[S])}{|S|^p}. \] We determine the sharp asymptotic upper bound for $λ(G)$ in terms of $d_p(G)$ and $n$. More precisely, every $n$-vertex graph $G$ with at least one edge satisfies \[ λ(G)\le \begin{cases} \left(\left(\max_{t\in\mathbb N_{\ge1}}\dfrac{t}{(t+1)^p}\right)^{-1}+o(1)\right)d_p(G)\sqrt n,&1<p<3/2,\\[0.4em] \left(\dfrac{3\sqrt3}{4}+o(1)\right)d_p(G)\sqrt{n\log n},&p=3/2,\\[0.4em] (\mathfrak C_p+o(1))d_p(G)n^{p-1},&3/2<p<2, \end{cases} \] and each constant here is best possible. Here $\mathfrak C_p$ is characterized by an exact variational problem over finite kernels. We apply a sparse graphon operator estimate to convert hereditary $p$-density bounds into sharp spectral bounds, and this estimate also explains the transition at the critical exponent $p=3/2$. For the endpoint $p=2$, Wilf's theorem gives the exact finite-$n$ bound $λ(G)\le 2d_2(G)n$, with equality for $K_n$. Thus Guiduli's power-law problem is resolved in its sharp asymptotic form for every $1<p\leq2$, including exact leading constants.

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arXiv2026

Polylogarithmic Bounds for Nested Cycles without Geometric Crossings

A problem of Erdős asks for extremal conditions forcing edge-disjoint cycles with a prescribed nested structure. In the geometric version, the nesting is required to be noncrossing with respect to the cyclic orders. Fernández, Kim, Kim and Liu proved that constant average degree forces two such cycles. We prove a polylogarithmic bound for the natural multi-layer version: for every fixed $k\ge 3$, every sufficiently large $n$-vertex graph with at least \[ C_k n(\log n)^{k-1}(\log\log n)^{k-3} \] edges contains $k$ pairwise edge-disjoint nested cycles without geometric crossings. The proof combines the robust sublinear expander framework of Alon, Bucić, Sauermann, Zakharov and Zamir with a controlled wrapping lemma that permits the layers to be built successively with controlled length.

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arXiv2026

On the Borodin--Kostochka conjecture for graphs with large maximum degree

The Borodin--Kostochka conjecture states that every graph $G$ with maximum degree $Δ(G)\ge 9$ satisfies $χ(G)\le \max\{ω(G),Δ(G)-1\}$. In this paper, we verify this conjecture for graphs with sufficiently large maximum degree. More precisely, we prove that every graph $G$ with maximum degree $Δ\ge 5.3\times 10^6$ and clique number $ω(G)<Δ$ satisfies $χ(G)\le Δ-1$. This improves a longstanding result of Reed.

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arXiv2026

A note on the $t$-partite link problem of Füredi

Motivated by the Erdős--Sós bipartite link conjecture, Füredi (Oberwolfach, 2004) asked for the asymptotic maximum edge density $π_{\mathrm{link}}(t)$ of $3$-graphs in which the link graph of every vertex is $t$-partite. Goldwasser's recursive blow-up construction based on projective planes gives the lower bound $π_{\mathrm{link}}(t)\ge 1-t^{-1}-(2+o_t(1))t^{-2}$ whenever $t-1$ is a prime power. In this note, we prove the upper bound $π_{\mathrm{link}}(t)\le 1-t^{-1}-t^{-2}/12$ for every $t \ge 2$. Together with Goldwasser's construction, this determines, up to a constant factor, the correct order of the gap between $π_{\mathrm{link}}(t)$ and the trivial averaging upper bound $1-t^{-1}$ for all prime-power values of $t-1$. In fact, our argument applies in the more general setting of $3$-graphs with no generalized daisies, equivalently, $3$-graphs in which the link graph of every vertex is $K_{t+1}$-free. We also establish an analogous upper bound for the positive $(r-1)$-codegree Turán density of generalized daisies.

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arXiv2026

On the $(k+2,k)$-problem of Brown, Erdős and Sós for even integers $k$

Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $r$-graph on $n$ vertices in which every $k$ edges span more than $s$ vertices. Brown, Erdős and Sós in 1973 conjectured that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists and verified the conjecture for $k=2$ by showing that $\lim_{n\to\infty} n^{-2} f^{(3)}(n;4,2)=\frac{1}{6}$. Delcourt and Postle, building on the work of Glock, Joos, Kim, Kühn, Lichev and Pikhurko, proved that for every $k\geq 2$, the limit $\lim_{n\to\infty} n^{-2} f^{(3)}(n;k+2,k)$ exists, thereby solving this conjecture. Their approach was later generalised by Shangguan to every uniformity $r\geq 4$: the limit $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ exists for all $r\geq 3$ and $k\geq 2$. However, its exact value was not determined. When $k\in\{2,3,\ldots,7\}$, the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n; rk-2k+2,k)$ were determined by Glock, Joos, Kim, Kühn, Lichev, Pikhurko, Rödl and Sun. Very recently, the limit for $k=8$ and $r\geq 4$ was determined by Pikhurko and Sun. For a general even integer $k$, Letzter and Sgueglia obtained the exact values of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k$ and uniformity $r\geq 2+\sqrt{2}\,k^{3/2}$. In this paper, we determine the exact value of $\lim_{n\to\infty} n^{-2} f^{(r)}(n;rk-2k+2,k)$ for every even integer $k\geq 4$ and $r\geq 2+\sqrt{\frac{3}{2}k-4}$, and show that it is $\frac{1}{r^2-r}.$

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arXiv2025

A hypergraph analogue of Alon-Frankl Theorem

Recently, Alon and Frankl (JCTB, 2024) determined the maximum number of edges in $K_{\ell+1}$-free $n$-vertex graphs with bounded matching number. For integers $\ell\ge r \ge 2$, the family $\mathcal{K}_{\ell+1}^{r}$ consists of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that, for some $(\ell+1)$-set $K$, every pair $\{x,y\} \subseteq K$ is covered by an edge in $F$. In this paper, we study the maximum number of edges in $\mathcal{K}_{\ell+1}^r$-free $r$-uniform hypergraphs that have the matching number at most $s$, that is, $\mathrm{ex}_r(n, \{\mathcal{K}_{\ell+1}^r, M^r_{s+1}\})$, and obtain the exact value for sufficiently large $n$, along with the corresponding extremal hypergraph. This result can be viewed as a hypergraph extension of the work of Alon and Frankl. In addition, for the $3$-uniform Fano plane $\mathbb{F}$, we determine the exact value of $\mathrm{ex}_3(n, \{\mathbb{F}, M^3_{s+1}\})$, and characterize the corresponding extremal hypergraph.

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arXiv2025

Generalized Turán results for disjoint copies of degenerate graphs

The generalized Turán number $\mathrm{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. For an integer $t \geq 1$, let $tF$ be the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku, and Vizer (2019) established an asymptotically sharp bound for $\mathrm{ex}(n,K_r,(t+1)K_{2,b})$. We extend their results in two directions by considering forbidden graphs $(t+1)K_{a,b}$ and $(t+1)C_{2k}$ and establish more precise matching upper and lower bounds of the same order of magnitude.

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arXiv2025

Generalized Turan number with given size

Generalized Turán problem with given size, denoted as $\mathrm{mex}(m,K_r,F)$, determines the maximum number of $K_r$-copies in an $F$-free graph with $m$ edges. We prove that for $r\ge 3$ and $α\in(\frac 2 r,1]$, any graph $G$ with $m$ edges and $Ω(m^{\frac{αr}{2}})$ $K_r$-copies has a subgraph of order $n_0=Ω(m^\fracα{2})$, which contains $Ω(n_0^{\frac{i(r-2)α}{(2-α)r-2}})$ $K_i$-copies for each $i = 2, \ldots, r$. This implies an upper bound of $\mathrm{mex}(m, K_r, F)$ when an upper bound of $\mathrm{ex}(n,K_r,F)$ is known. Furthermore, we establish an improved upper bound of $\mathrm{mex}(m, K_r, F)$ by $\mathrm{ex}(n, F)$ and $\min_{v_0 \in V(F)} \mathrm{ex}(n, K_r, F - v_0)$. As a corollary, we show $\mathrm{mex}(m, K_r, K_{s,t}) = Θ( m^{\frac{rs - \binom{r}{2}}{2s-1}} )$ for $r \geq 3$, $s \geq 2r-2$ and $t \geq (s-1)! + 1$, and obtain non-trivial bounds for other graph classes such as complete $r$-partite graphs and $K_s \vee C_\ell$, etc.

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