Multicolor Ramsey numbers for triple systems

Given an r -uniform hypergraph H , the multicolor Ramsey number r k ( H ) is the minimum n such that every k -coloring of the edges of the complete r -uniform hypergraph K n r yields a monochromatic copy of H . We investigate r k ( H ) when k grows and H is fixed. For nontrivial 3 -uniform hypergraphs H , the function r k ( H ) ranges from 6 k ( 1 + o ( 1 ) ) to double exponential in k . erve that r k ( 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 r with s ≥ s 0 ( 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. -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 r k ( K 3 ) ≤ r 4 k ( K 4 3 − e ) ≤ r 4 k ( K 3 ) + 1 , where K 4 3 − e is obtained from K 4 3 by deleting an edge. vide some other bounds, including single-exponential bounds for F 5 = { a b e , a b d , c d e } as well as asymptotic or exact values of r k ( H ) when H is the bow { a b c , a d e } , kite { a b c , a b d } , tight path { a b c , b c d , c d e } or the windmill { a b c , b d e , c e f , b c e } . We also determine many new “small” Ramsey numbers and show their relations to designs. For example, the lower bound for r 6 ( k i t e ) = 8 is demonstrated by decomposing the triples of a seven element set into six partial STS (two of them are Fano planes).