组合数学

A graph is non-$r$-partite if its chromatic number exceeds $r$. For an edge-color-critical graph $F$ with $\chi(F)=r+1$, let $\mathrm{ex}_{r+1,\rho}(n,F)$ be the maximum adjacency spectral radius among non-$r$-partite $F$-free graphs of…

Let $\mathcal Q=\{Q_a:a\geq1\}$ be a nested family of finite posets such that $Q_a\subseteq Q_{a+1}$ and $|Q_a|<|Q_{a+1}|$. For a poset $Q$, let $\mathcal C_t(Q)$ denote the set of all strict $t$-chains in $Q$. Given an $r$-coloring of…

We study finite-field analogues of the Peres--Schlag nonempty-interior problem for product sets. Given \(A\subseteq\mathbb F_p\), we ask when a suitable one-dimensional linear image of \(A^n\) is full; equivalently, when there exist…

A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements. We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size…

Let $p$ be a binary word of length $\ell$ with $r\geq2$ runs. Previously known only for $k\leq4$, we show for $n$ sufficiently large that the number of binary words of length $n$ with exactly $k$ subsequences equal to $p$ is polynomial in…

Unit-interval parking functions of length $n$ are enumerated by the Fubini numbers $F_n$ and are in explicit bijection with the ordered set partitions of $[n]$. We use this bijection to single out the unit-interval parking functions whose…

The Chromatic Sum problem asks, given a graph $G$ and an integer $k$, whether $G$ admits a colouring $c$ with sum $\sum_{v\in V}c(v) \leq k$. We study the complexity of Chromatic Sum on graph classes defined by some set of forbidden graphs.…

Let $f\colon{\mathbb Z}^2\to{\mathbb Z}$ be a Riemann function whose weight $W$ is a perfect matching. Then there is a family of sheaves of $k$-vector spaces $\{{{M}}_{W,{\bf d}}\}_{{\bf d}\in{\mathbb Z}^2}$ on a five-point topological that…

A strong majority edge-coloring of a graph is an edge-coloring in which, for every edge $e$ and every color $i$, at most half of the edges adjacent to $e$ have color $i$. Such a coloring exists only for graphs with no pendant path of length…

The Boolean lattice $BL_n$, $n\geq 3$, is the graph whose vertex set is the collection of all subsets of $[n]=\{1,2,\ldots,n\}$, where two subsets $U$ and $W$ are adjacent if and only if their symmetric difference has precisely one element.…

We prove the sharp diagonal spectral correlation conjecture of Friedgut, Kahn, Kalai and Keller, proposed in their Fourier-analytic approach to Chv\'atal's conjecture. For every pair of increasing Boolean functions…

Let $k_r(G)$ denote the number of $r$-cliques in a graph $G$ and let $F_r(\cdot)$ be the Lov\'asz--Simonovits $r$-clique density function. For any integers $2\le s<t$, we determine the asymptotically sharp lower bound on $k_t(G)$ in an…

The familiar color wheel is a disk divided into six sectors, colored red, orange, yellow, green, blue, and purple, in circular order. Three of the colors can be obtained by blending the colors in the two neighboring sectors. One might…

List-coloring, introduced independently by Vizing and by Erd\H{o}s, Rubin, and Taylor in the 1970s, generalizes ordinary vertex coloring by assigning to each vertex its own set of admissible colors. A graph is chromatic-choosable if its…

Ziegler proved that every simplicial $d$-dimensional $0/1$-polytope has at most $2d$ vertices, and asked whether equality forces the polytope to be centrally symmetric and hence, equivalently, a $0/1$-realization of the $d$-dimensional…

The feedback arc set problem on tournaments arises in a rich variety of applications, and has been studied extensively in several research fields over the past six decades. It is well known that this problem is $NP$-hard and admits a…

We report the exact value of the number of labeled partially ordered sets (equivalently, labeled $T_0$ topologies) on 19 points, P(19) = 646099441937791106493755218560442089979, a 39-digit integer extending OEIS A001035, whose largest…

Partial cubes are a fundamental class of graphs that admit isometric embeddings into hypercubes. Klav\v{z}ar and Kov\v{s}e [Ars Combin. 93 (2009), 77--86] observed that the opposite semicubes of every harmonic-even partial cube are pairwise…

A rainbow stacking of $m$ independent, uniformly random $r$-edge-colourings of $K_n$ is a tuple of vertex permutations that superimposes the colourings such that no two edges of the same colour overlap. The study of the critical palette…

A total rainbow forest in an edge-colored graph is a forest that contains every edge color exactly once. Using a necessary and sufficient condition that a total rainbow forest exists, we demonstrate the existence of huge numbers of…