Abstract
Using a $\mathbb{Z}_q$-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 uniform hypergraph).
Highlights
1.1 Motivations and resultsA hypergraph is a pair H = (V (H), E(H)), where V (H) is a finite set and E(H) a family of subsets of V (H)
Using a Zq-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
Summary
The chromatic number of such a hypergraph, denoted χ(H), is the minimal value of t for which a proper coloring exists. The 2-colorability defect cd2(H) of a hypergraph H has been introduced by Dol’nikov [3] in 1988 for a generalization of Lovasz’s theorem. It is defined as the minimum number of vertices that must be removed from H so that the hypergraph induced by the remaining vertices is of chromatic number at most 2: cd2(H) = min{|Y | : Y ⊆ V (H), χ(H[V (H) \ Y ]) 2}. Our main result is the following extension of Simonyi-Tardos’s theorem to Kneser hypergraphs. Whether Theorem 1 is true for non-prime p is an open question
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