Colorful Subhypergraphs in Uniform Hypergraphs

There are several topological results ensuring in any properly colored graph the existence of a colorful complete bipartite subgraph, whose order is bounded from below by some topological invariants of some topological spaces associated to the graph. Meunier [Electron. J. Combin., 2014] presented the first colorful type result for uniform hypergraphs. In this paper, we give some new generalizations of the Zp-Tucker Lemma and by use of them, we improve Meunier’s result and some other colorful results by Simonyi, Tardif, and Zsbán [Electron. J. Combin., 2014] and by Simonyi and Tardos [Electron. J. Combin., 2007] to uniform hypergraphs. Also, we introduce some new lower bounds for the chromatic number and local chromatic number of uniform hypergraphs. A hierarchy between these lower bounds is presented as well.