  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.
