New bounds of two hypergraph Ramsey problems

TL;DR

This paper advances the understanding of hypergraph Ramsey problems by establishing a superexponential lower bound for a specific Erdős-Hajnal function and improving upper bounds for the Erdős-Rogers function, thus making progress on longstanding conjectures.

Contribution

It proves a superexponential lower bound for $r_4(5,4;n)$ and refines upper bounds for the hypergraph Erdős-Rogers function, advancing the field's knowledge on hypergraph Ramsey problems.

Findings

Established a superexponential lower bound for $r_4(5,4;n)$.

Improved the upper bound for the hypergraph Erdős-Rogers function to an iterated $(k-3)$-fold logarithm.

Progressed towards resolving conjectures on the growth rates of hypergraph Ramsey functions.

Abstract

We focus on two hypergraph Ramsey problems. First, we consider the Erd\H{o}s-Hajnal function $r_k(k+1,t;n)$. In 1972, Erd\H{o}s and Hajnal conjectured that the tower growth rate of $r_k(k+1,t;n)$ is $t-1$ for each $2\le t\le k$. To finish this conjecture, it remains to show that the tower growth rate of $r_4(5,4;n)$ is three. We prove a superexponential lower bound for $r_4(5,4;n)$, which improves the previous best lower bound $r_4(5,4;n)\geq 2^{\Omega(n^2)}$ from Mubayi and Suk (\emph{J. Eur. Math. Soc., 2020}). Second, we prove an upper bound for the hypergraph Erd\H{o}s-Rogers function $f^{(k)}_{k+1,k+2}(N)$ that is an iterated $(k-3)$-fold logarithm in $N$ for each $k\geq 5$. This improves the previous upper bound that is an iterated $(k-13)$-fold logarithm in $N$ for $k\ge14$ due to Mubayi and Suk (\emph{J. London Math. Soc., 2018}), in which they conjectured that $f^{(k)}_{k+1,k+2}(N)$ is an iterated $(k-2)$-fold logarithm in $N$ for each $k\ge3$.