Off-Diagonal Ramsey Numbers for Linear Hypergraphs

TL;DR

This paper investigates off-diagonal Ramsey numbers for linear hypergraphs, revealing that for uniformity $k \\ge 4$, these numbers can grow extremely rapidly, much faster than previously conjectured for the case $k=3$.

Contribution

The paper demonstrates that in higher uniformities, off-diagonal Ramsey numbers can grow at tower-exponential rates, contrasting with earlier polynomial growth expectations.

Findings

For $k \\ge 4$, constructed hypergraphs with super-exponential Ramsey numbers.

Disproved the polynomial growth conjecture for $k=3$ in earlier work.

Established lower bounds showing extremely rapid growth in higher uniformities.

Abstract

We study off-diagonal Ramsey numbers $r(H, K_n^{(k)})$ of $k$-uniform hypergraphs, where $H$ is a fixed linear $k$-uniform hypergraph and $K_n^{(k)}$ is complete on $n$ vertices. Recently, Conlon et al.\ disproved the folklore conjecture that $r(H, K_n^{(3)})$ always grows polynomially in $n$. In this paper we show that much larger growth rates are possible in higher uniformity. In uniformity $k\ge 4$, we prove that for any constant $C>0$, there exists a linear $k$-uniform hypergraph $H$ for which $$r(H,K_n^{(k)}) \geq \textup{twr}_{k-2}(2^{(\log n)^C}).$$