Hypergraphs with many Kneser colorings

For fixed positive integers r, k and ` with 1 ≤ ` < r and an r-uniform hypergraph H, let κ(H, k, `) 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 ` elements. Consider the function KC(n, r, k, `) = maxH∈Hn κ(H, k, `), 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, `) for every fixed r, k and ` 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 [7] on intersecting systems of sets.