The chromatic number of Kneser hypergraphs (original) (raw)

Suppose the r-subsets of an n-element set are colored by t colors. THEOREM 1.1. If n > (t-1)(k-1) + k * r, then there are k pairwise disjoint r-sets having the same color. This was conjectured by Erd6s [E] in 1973. Let T(n, r, s) denote the Turin number for s-uniform hypergraphs (see ?1). THEOREM 1.3. If e > 0, t < (1-e)T(n, r, s)/(k-1), and n > no(e, r, s, k), then there are k r-sets A1,A2,...,Ak having the same color such that Ai nfAjI < s for all 1 < i < j < k. If s = 2, e can be omitted.-Theorem 1.1 is best possible. Its proof generalizes Lov6sz' topological proof of the Kneser conjecture (which is the case k = 2). The proof uses a generalization, due to Bariny, Shlosman, and Sziics of the Borsuk-Ulam theorem. Theorem 1.3 is best possible up to the c-term (for large n). Its proof is purely combinatorial, and employs results on kernels of sunflowers.