# New lower bounds for hypergraph Ramsey numbers

## TL;DR

This paper establishes significantly improved lower bounds for 4-uniform hypergraph Ramsey numbers, demonstrating their growth is double exponential in a power of n, advancing understanding of their asymptotic behavior.

## Contribution

The paper introduces new lower bounds for specific 4-uniform hypergraph Ramsey numbers, nearly resolving the tower growth rate question for all classical off-diagonal hypergraph Ramsey numbers.

## Key findings

- Proves $r_4(5,n) > 2^{n^{c\log n}}$
- Establishes $r_4(6,n) > 2^{2^{cn^{1/5}}}$
- Shows growth rate of $r_4(6,n)$ is double exponential in a power of n

## Abstract

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: $$r_4(5,n) > 2^{n^{c\log n}} \qquad \hbox{ and } \qquad r_4(6,n) > 2^{2^{cn^{1/5}}},$$ where $c$ is an absolute positive constant. This substantially improves the previous best bounds of $2^{n^{c\log\log n}}$ and $2^{n^{c\log n}}$, respectively. Using previously known upper bounds, our result implies that the growth rate of $r_4(6,n)$ is double exponential in a power of $n$. As a consequence, we obtain similar bounds for the $k$-uniform Ramsey numbers $r_k(k+1, n)$ and $r_k(k+2, n)$ where the exponent is replaced by an appropriate tower function. This almost solves the question of determining the tower growth rate for {\emph {all}} classical off-diagonal hypergraph Ramsey numbers, a question first posed by Erd\H os and Hajnal in 1972. The only problem that remains is to prove that $r_4(5,n)$ is double exponential in a power of $n$.