A celebrated result in Ramsey Theory states that the order of magnitude of the triangle-complete graph Ramsey numbers is . In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges such that and . For all , let be the smallest positive integer n such that in every red–blue coloring of the edges of the complete r-uniform hypergraph , there exists a red triangle or a blue . We show that there exist constants such that for all , and for This determines up to a logarithmic factor the order of magnitude of . We conjecture that for all . We also study a generalization to hypergraphs of cycle-complete graph Ramsey numbers and a connection to , the maximum size of a set of integers in not containing a three-term arithmetic progression.
Research of this author is supported in part by NSF grant DMS-0965587 and by grants 12-01-00448-a and 12-01-00631 of the Russian Foundation for Basic Research.