Colorful Subhypergraphs in Uniform Hypergraphs
  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. 
Meysam Alishahi
math.CO