Hypergraph Ramsey numbers: Triangles versus cliques

A celebrated result in Ramsey Theory states that the order of magnitude of the trianglecomplete graph Ramsey numbers R(3, t) is t/ log t. In this paper, we consider an analogue of this problem for uniform hypergraphs. A triangle is a hypergraph consisting of edges e, f, g such that |e∩ f | = |f ∩ g| = |g ∩ e| = 1 and e ∩ f ∩ g = ∅. For all r ≥ 2, let R(C3,K t ) be the smallest positive integer n such that in every red-blue coloring of the edges of the complete r-uniform hypergraph K n, there exists a red triangle or a blue K r t . We show that there exist constants a, br > 0 such that for all t ≥ 3, at 3 2 (log t) 3 4 ≤ R(C3,K t ) ≤ b3t 3 2