Lower bounds for Ramsey numbers of bounded degree hypergraphs

TL;DR

This paper establishes new lower bounds on the 4-color Ramsey numbers for bounded degree hypergraphs, extending previous graph results and answering a longstanding open question.

Contribution

It proves that for all uniformities k ≥ 3, there exist hypergraphs with large Ramsey numbers matching tower function bounds, extending known graph results to hypergraphs.

Findings

Ramsey number lower bounds grow as tower functions of degree and parameters

Bounds are tight for k ≥ 4, nearly tight for k=3

Answers a 2008 open question in hypergraph Ramsey theory

Abstract

We prove that, for all $k \ge 3,$ and any integers $\Delta, n$ with $n \ge \Delta,$ there exists a $k$-uniform hypergraph on $n$ vertices with maximum degree at most $\Delta$ whose $4$-color Ramsey number is at least $\mathrm{tw}_k(c_k \Delta) \cdot n$, for some constant $c_k > 0$, where $\mathrm{tw}_k$ denotes the tower function. For $k \ge 4,$ this is tight up to the constant $c_k$ and for $k = 3$ it is known to be tight up to a factor of $\log \Delta$ on top of the tower. It extends a well-known result of Graham, R\"{o}dl and Ruci\'{n}ski for graphs and answers a question of Conlon, Fox and Sudakov from 2008.