Lower bounds for Ramsey numbers of bounded degree hypergraphs

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.