Multicolor Ramsey numbers for triple systems

Given an r-uniform hypergraph H, the multicolor Ramsey number rk(H) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph K n yields a monochromatic copy of H. We investigate rk(H) when k grows and H is fixed. For nontrivial 3-uniform hypergraphs H, the function rk(H) ranges from √ 6k(1 + o(1)) to double exponential in k. We observe that rk(H) is polynomial in k when H is r-partite and at least single-exponential in k otherwise. Erdős, Hajnal and Rado gave bounds for large cliques K s with s ≥ s0(r), showing its correct exponential tower growth. We give a proof for cliques of all sizes, s > r, using a slight modification of the celebrated stepping-up lemma of Erdős and Hajnal. For 3-uniform hypergraphs, we give an infinite family with sub-double-exponential upper bound and show connections between graph and hypergraph Ramsey numbers. Specifically, we prove that rk(K3) ≤ r4k(K 4 − e) ≤ r4k(K3) + 1, where K 4 − e is obtained from K 4 by deleting an edge. We provide some other bounds, including single-exponential bounds for F5 = {abe, abd, cde} as well as asymptotic or exact values of rk(H) when H is the bow {abc, ade}, kite {abc, abd}, tight path {abc, bcd, cde} or the windmill {abc, bde, cef, bce}. We also determine many new “small” Ramsey numbers and show their relations to designs. For example, the lower bound for r6(kite) = 8 is demonstrated by decomposing the triples of a seven element set into six partial STS (two of them are Fano planes).