相关论文: On generalized Kneser hypergraph colorings
The notion of $S$-labeling of graphs, where $S$ is a subset of a symmetric group, was introduced in 2019 by Jin, Wong, and Zhu. This notion provides the framework for a common generalization of various well studied notions of graph…
The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set…
We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $\Omega(\sqrt{n})$. This improves on the…
A graph $H$ is common if its Ramsey multiplicity, i.e., the minimum number of monochromatic copies of $H$ contained in any $2$-edge-coloring of $K_n$, is asymptotically the same as the number of monochromatic copies in the random…
The $c$-strong chromatic number of a hypergraph is the smallest number of colours needed to colour its vertices so that every edge sees at least $c$ colours or is rainbow. We show that every $t$-intersecting hypergraph has bounded $(t +…
One formulation of the Erdos-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on $\{1, 2, \ldots, N\}$, there exists a monochromatic red $s$-clique or a monochromatic blue…
By Lovasz' proof of the Kneser conjecture, the chromatic number of a graph G is bounded from below by the index of the Z_2-space Hom(K_2,G) plus two. We show that the cohomological index of Hom(K_2,G) is also greater than the cohomological…
This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H.…
In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that…
The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of $b$-simple hypergraphs, in which any two edges do not share more than $b$ common vertices. We prove that for…
Computing cohesive subgraphs is a central problem in graph theory. While many formulations of cohesive subgraphs lead to NP-hard problems, finding a densest subgraph can be done in polynomial time. As such, the densest subgraph model has…
A classical result of Erd\H{o}s and Hajnal claims that for any integers $k, r, g \geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ with chromatic number at least $k$. This implies that there are sparse hypergraphs such that…
The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of…
Hadwiger's conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_t$ as a minor is properly $(t-1)$-colorable. The purpose of this work is to demonstrate that a natural extension of…
A well-known conjecture, often attributed to Ryser, states that the cover number of an $r$-partite $r$-uniform hypergraph is at most $r - 1$ times larger than its matching number. Despite considerable effort, particularly in the…
A famous conjecture of Ryser states that every $r$-partite hypergraph has vertex cover number at most $r - 1$ times the matching number. In recent years, hypergraphs meeting this conjectured bound, known as $r$-Ryser hypergraphs, have been…
A constrained colouring or, more specifically, an $(\alpha,\beta)$-colouring of a hypergraph $H$, is an assignment of colours to its vertices such that no edge of $H$ contains less than $\alpha$ or more than $\beta$ vertices with different…
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions.…
A (finite, undirected) graph is $(n,k)$-colourable if we can assign each vertex a $k$-subset of $\{1,2,\ldots,n\}$ so that adjacent vertices receive disjoint subsets. We consider the following problem: if a graph is $(n,k)$-colourable, then…
Given a graph $G$ and an integer $p$, a coloring $f : V(G) \to \mathbb{N}$ is \emph{$p$-centered} if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$.…