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 Erdo s and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdo s-Ko-Rado Theorem (Erdo s etal., 1961. ) on intersecting systems of sets. © 2011 Elsevier Ltd.
Hoppen, C., Kohayakawa, Y., & Lefmann, H. (2012). Hypergraphs with many Kneser colorings. European Journal of Combinatorics, 33(5), 816–843. https://doi.org/10.1016/j.ejc.2011.09.025