On Chromatic Number of Kneser Hypergraphs

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.