Hypergraph Ramsey numbers: Triangles versus cliques

A celebrated result in Ramsey Theory states that the order of magnitude of the triangle-complete graph Ramsey numbers R ( 3 , t ) is t 2 / 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 ( C 3 , K t r ) be the smallest positive integer n such that in every red–blue coloring of the edges of the complete r-uniform hypergraph K n r , there exists a red triangle or a blue K t r . We show that there exist constants a , b r > 0 such that for all t ⩾ 3 , a t 3 2 ( log t ) 3 4 ⩽ R ( C 3 , K t 3 ) ⩽ b 3 t 3 2 and for r ⩾ 4 t 3 2 ( log t ) 3 4 + o ( 1 ) ⩽ R ( C 3 , K t r ) ⩽ b r t 3 2 . This determines up to a logarithmic factor the order of magnitude of R ( C 3 , K t r ) . We conjecture that R ( C 3 , K t r ) = o ( t 3 / 2 ) for all r ⩾ 3 . We also study a generalization to hypergraphs of cycle-complete graph Ramsey numbers R ( C k , K t ) and a connection to r 3 ( N ) , the maximum size of a set of integers in { 1 , 2 , … , N } not containing a three-term arithmetic progression.