On Chromatic Number of Kneser Hypergraphs

TL;DR

This paper introduces new lower bounds for the chromatic number of Kneser hypergraphs using the $Z_p$-Tucker lemma, determines the chromatic number for certain multiple Kneser hypergraphs, and extends results on almost $s$-stable Kneser hypergraphs, also providing a colorful-type theorem.

Contribution

It improves existing bounds on Kneser hypergraph chromatic numbers, specifies the chromatic number for new classes of multiple Kneser hypergraphs, and extends stability results to broader hypergraph families.

Findings

Introduces a lower bound for chromatic number using $Z_p$-Tucker lemma.

Determines chromatic number of certain multiple Kneser hypergraphs.

Extends stable Kneser hypergraph results to larger classes.

Abstract

In this paper, in view of $Z_p$-Tucker lemma, we introduce a lower bound for chromatic number of Kneser hypergraphs which improves Dol'nikov-K{\v{r}}{\'{\i}}{\v{z}} bound. Next, we introduce multiple Kneser hypergraphs and we specify the chromatic number of some multiple Kneser hypergraphs. For a vector of positive integers $\vec{s}=(s_1,s_2,\ldots,s_m)$ and a partition $\pi=(P_1,P_2,\ldots,P_m)$ of $\{1,2,\ldots,n\}$, the multiple Kneser hypergraph ${\rm KG}^r(\pi; \vec{s};k)$ is a hypergraph with the vertex set $$V=\left\{A:\ A\subseteq P_1\cup P_2\cup\cdots \cup P_m,\ |A|=k, \forall 1\leq i\leq m;\ |A\cap P_i|\leq s_i\right\}$$ whose edge set is consist of any $r$ pairwise disjoint vertices. We determine the chromatic number of multiple Kneser hypergraphs provided that $r=2$ or for any $1\leq i\leq m$, we have $|P_i|\leq 2s_i$. A subset $S \subseteq [n]$ is almost $s$-stable if for any two distinct elements $i,j\in S$, we have $|i-j|\geq s$. The almost $s$-stable Kneser hypergraph ${\rm KG}^r(n,k)_{s-stab}^{\sim}$ has all $s$-stable subsets of $[n]$ as the vertex set and every $r$-tuple of pairwise disjoint vertices forms an edge. Meunier [The chromatic number of almost stable Kneser hypergraphs. J. Combin. Theory Ser. A, 118(6):1820--1828, 2011] showed for any positive integer $r$, $\chi({\rm KG}^r(n,k)_{2-stab}^{\sim})=\left\lceil {n-r(k-1) \over r-1}\right\rceil$. We extend this result to a large family of Schrijver hypergraphs. Finally, we present a colorful-type result which confirms the existence of a completely multicolored complete bipartite graph in any coloring of a graph.