Hypergraphs with many Kneser colorings

Abstract

For fixed positive integers r,k and ℓ with 1≤ℓ<r and an r-uniform hypergraph H, let κ(H,k,ℓ) denote the number of k-colorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least ℓ elements. Consider the function KC(n,r,k,ℓ)=maxH∈Hnκ(H,k,ℓ), where the maximum runs over the family Hn of all r-uniform hypergraphs on n vertices. In this paper, we determine the asymptotic behavior of the function KC(n,r,k,ℓ) for every fixed r, k and ℓ and describe the extremal hypergraphs. This variant of a problem of Erdős and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdős–Ko–Rado Theorem (Erdős et al., 1961 [8]) on intersecting systems of sets.