Colorful Subhypergraphs in Uniform Hypergraphs

TL;DR

This paper generalizes topological methods to establish new bounds on the chromatic and local chromatic numbers of uniform hypergraphs, extending previous results on colorful subhypergraphs in Kneser hypergraphs.

Contribution

It introduces new generalizations of the $ ext{Z}_p$-Tucker lemma, improving upon Meunier's results and providing novel lower bounds for hypergraph chromatic numbers.

Findings

New lower bounds for chromatic numbers of uniform hypergraphs

Enhanced results on colorful subhypergraphs in Kneser hypergraphs

Generalizations of the $ ext{Z}_p$-Tucker lemma

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 ${\rm KG}^r({\mathcal H})$. 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 ${\rm KG}^r({\mathcal H})$ 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.