Hypergraphs with many Kneser colorings

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 ) = max H ∈ H n κ ( H , k , ℓ ) , where the maximum runs over the family H n 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.