Kneser Colorings of Uniform Hypergraphs

For fixed positive integers r, k and l with 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 vertices. 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 ) and describe the extremal hypergraphs. This variant of a problem of Erdős and Rothschild, who considered colorings of graphs without a monochromatic triangle, is related to the Erdős-Ko-Rado Theorem [Erdős, P., C. Ko, and R. Rado, Intersection theorems for systems of finite sets, Quarterly Journal of Mathematics, Oxford Series, series 2, 12 (1961), 313–320] on intersecting systems of sets.