Multicolor Ramsey numbers for triple systems [article]

Maria Axenovich, Andras Gyarfas, Hong Liu, Dhruv Mubayi
2013 arXiv   pre-print
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 √(6k)(1+o(1)) to double exponential in k. We observe that r_k(H) is polynomial in k when H is r-partite and at least single-exponential in k otherwise. Erdős, Hajnal and
more » ... 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. 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 r_k(K_3)< r_4k(K_4^3-e)< r_4k(K_3)+1, where K_4^3-e is obtained from K_4^3 by deleting an edge. We provide some other bounds, including single-exponential bounds for F_5={abe,abd,cde} as well as asymptotic or exact values of r_k(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 r_6(kite)=8 is demonstrated by decomposing the triples of [7] into six partial STS (two of them are Fano planes).
arXiv:1302.5304v1 fatcat:3gihbc3webai5koy7sjwjdu5r4