Multicolor Ramsey numbers for triple systems

Abstract

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 Knr 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 Ksr 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 thatrk(K3)≤r4k(K43−e)≤r4k(K3)+1, where K43−e is obtained from K43 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).