Colorful hypergraphs in Kneser hypergraphs

TL;DR

This paper generalizes a theorem to Kneser hypergraphs, establishing conditions for multicolored bipartite subgraphs and deriving bounds for their local chromatic number, advancing understanding of hypergraph colorings.

Contribution

It extends Ky Fan's theorem to Kneser hypergraphs, providing new bounds on their local chromatic number and revealing multicolored bipartite structures.

Findings

Established a lower bound for the local chromatic number of Kneser hypergraphs

Proved the existence of multicolored complete bipartite subgraphs in proper colorings

Generalized Ky Fan's theorem using a $Z_q$-framework

Abstract

Using a $Z_q$-generalization of a theorem of Ky Fan, we extend to Kneser hypergraphs a theorem of Simonyi and Tardos that ensures the existence of multicolored complete bipartite graphs in any proper coloring of a Kneser graph. It allows to derive a lower bound for the local chromatic number of Kneser hypergraphs (using a natural definition of what can be the local chromatic number of a hypergraph).