组合数学

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family, $\Delta(\mathcal{F})=\max_{x\in[n]}|\{F\in\mathcal{F}:x\in F\}|$, and $\varrho(\mathcal{F})=\Delta(\mathcal{F})/|\mathcal{F}|$. Frankl and Wang conjectured that if $n>100k$…

We prove a sharp lower bound for the cardinality of sumsets of subsets of $\mathbb{Z}^d$ confined to a hypercube, resolving in strong form a conjecture that was made explicit by Becker, Ivanisvili, Krachun and Madrid and had circulated in…

Let $G$ be a finite simple graph. The annihilation number $a(G)$ is an efficiently computable upper bound on the independence number $\alpha(G)$. We develop a sharp matching-number theory for the gap $a(G)-\alpha(G)$. The strongest general…

Maker-Breaker subgraph games are among the most famous combinatorial games. For $n,q\in\mathbb{N}$ and a fixed subgraph $C$ of the complete graph $K_n$, the two players, called Maker and Breaker, alternately claim edges of $K_n$. Maker…

The greedy Tamari poset, inspired by the well-studied Tamari lattice, was recently defined by Dermenjian in the more general setting of greedy $\nu$-Tamari posets. Bousquet-M\'elou and Chapoton counted intervals of the greedy $m$-Tamari…

Let $F(N)$ denote the largest cardinality of a Sidon subset of $\{0, 1, \dots, N - 1\}$. We prove \[ F(N) \le N^{1/2} + 0.94601 N^{1/4} + O(1). \] This improves the recently announced coefficient $0.97633$ obtained by Carter, Georgiev,…

Very recently, using Meshulam's lemma, Blagojevi\'c proved a constrained version of the colorful Carath\'eodory theorem for joins of bipartite spanning trees and wedge of spheres. Our main contribution extends his result from joins of…

In this paper, we investigate the combinatorial structure arising from the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$, and $Z_p$, satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$,…

The generalized honeymoon Oberwolfach problem (HOP) asks whether it is possible to seat $2n$ participants consisting of $n$ newlywed couples at a conference with $s$ tables of size $2$ and $t$ "round'' tables of sizes $2m_1, 2m_2, \ldots,…

For any graph $G = (V,E)$, a subset $S {\subseteq} V$ dominates $G$ if $N[S] = V$. The minimum cardinality over all such $S$ is called the domination number, written ${\gamma}(G)$. The classical conjecture of V.G. Vizing states that…

We study the supersaturation problem in its edge-spectral form. Let $\lambda(G)$ be the adjacency spectral radius of $G$. Nikiforov proved that every $K_{r+1}$-free graph $G$ with $m$ edges satisfies $\lambda (G)\le \sqrt{(1\!-\!1/r )2m}$.…

Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $\mu,\lambda\vdash n$, the polynomial $(-1)^{|\mu|-\ell(\mu)}\langle \nabla m_\mu, s_\lambda\rangle$ has nonnegative integer coefficients, where $\nabla$ is…

We resolve (for all sufficiently large $n$) a conjecture of Pilz on the symmetric difference $A\Delta (2A)\Delta \cdots\Delta (nA)$ for finite sets $A\subseteq \mathbb{N}$ of positive integers. We show that this set always has cardinality…

Order polytopes for generalized snake posets were recently studied by von Bell et al. (2022), and are known to be unimodularly equivalent to strength-one flow polytopes for acyclic directed graphs strongly dual to generalized snake posets.…

We introduce the lower and upper Wythoff-Fibonacci sequences, obtained from the classical Wythoff sequences by a Fibonacci correction. Specifically, if we put $$\epsilon(j)=\begin{cases}(-1)^k, & \text{if }j=F_k\text{ for some }k\\ 0, &…

Let $\boldsymbol{a}=(a_i)_{i=1}^\infty$ be an infinite sequence of points on a circle. The first $n$ of these points cuts the circle into $n$ pieces. For any given $r$, let $\mu^r_n(\boldsymbol{a})$ be the ratio between the maximum and…

A graph $G$ is called $k$-edge hamiltonian if every linear forest (i.e., a disjoint union of paths) with at most $k$ edges is contained in a Hamilton cycle of $G$. Motivated by earlier results of Erd\H{o}s, F\"{o}redi, Kostochka and Luo…

Let \(F\) and \(G\) be \(r\)-uniform hypergraphs, and let \(f_{F,G}(n)\) be the largest integer \(m\) such that every \(n\)-vertex \(G\)-free \(r\)-graph contains an induced \(F\)-free subgraph on \(m\) vertices. We prove that, if…

A graph is balanced if its clique-matrix contains no square submatrix of odd order with exactly two $1$'s in each row and in each column. Although it is known that a graph is balanced if and only if it contains no induced extended odd sun,…

We extend the analysis of nonrepetitive sequences of Entringer et al. [Journal of Combinatorial Theory, 1974] to relaxations of equality testing under nonstandard equivalence relations, in particular parameterized equivalence and…