A double-exponential lower bound for $r_4(5,n)$

TL;DR

This paper establishes a double-exponential lower bound for the hypergraph Ramsey number r_4(5,n), solving a long-standing problem about the growth rate of off-diagonal hypergraph Ramsey numbers.

Contribution

It proves a new lower bound for r_4(5,n) and determines the tower growth rate of r_k(k+1,n), resolving a problem posed by Erdős and Hajnal in 1972.

Findings

r_4(5,n) ≥ 2^{2^{cn^{1/7}}} for some constant c

The tower growth rate of r_k(k+1,n) is now fully characterized

Solved the problem of growth rates for all classical off-diagonal hypergraph Ramsey numbers

Abstract

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute constant. As a consequence, we determine the tower growth rate of $r_k(k+1,n)$, which completely solves the problem of establishing the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers, first posed by Erd\H{o}s and Hajnal in 1972.