Colorful Subhypergraphs in Uniform Hypergraphs

Abstract

There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of ${\mathbb{Z}}_p$-Tucker lemma, Alishahi and Hajiabolhassan [{\it On the chromatic number of general Kneser hypergraphs, Journal of Combinatorial Theory, Series B, 2015}] introduced a lower bound for the chromatic number of Kneser hypergraphs ${\mathrm{KG}}^r(\mathcal{H})$. Next, Meunier [{\it Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014}] improved their result by proving that any properly colored general Kneser hypergraph ${\mathrm{KG}}^r(\mathcal{H})$ contains a large colorful $r$-partite subhypergraph provided that $r$ is prime. In this paper, we give some new generalizations of ${\mathbb{Z}}_p$-Tucker lemma. Hence, improving Meunier's result in some aspects. Some new lower bounds for the chromatic number and local chromatic number of uniform hypergraphs are presented as well.

18 pages