When are off-diagonal hypergraph Ramsey numbers polynomial?

Abstract

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

Theorems & Definitions (18)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• proof : Proof of Theorem \ref{['thm:all-tight']}
• Theorem 3.1
• Lemma 3.2
• proof : Proof of Theorem \ref{['thm:all-two-components']}
• Lemma 3.3
• proof