When are off-diagonal hypergraph Ramsey numbers polynomial?

Abstract

A natural open problem in Ramsey theory is to determine those 3 3 -graphs H H for which the off-diagonal Ramsey number r ( H , K n ( 3 ) ) r(H, K_n^{(3)}) grows polynomially with n n . We make substantial progress on this question by showing that if H H is tightly connected or has at most two tight components, then r ( H , K n ( 3 ) ) r(H, K_n^{(3)}) grows polynomially if and only if H H is contained in an iterated blowup of an edge.