Tower Gaps in Multicolour Ramsey Numbers

TL;DR

This paper constructs hypergraphs demonstrating large tower height differences between 2-colour and q-colour Ramsey numbers, introducing a new variant of the stepping-up lemma to establish tower-type lower bounds.

Contribution

It introduces a new variant of the Erdős–Hajnal stepping-up lemma for generalized Ramsey numbers, enabling the construction of hypergraphs with large tower gaps.

Findings

Hypergraphs with arbitrarily large tower height separation between 2-colour and q-colour Ramsey numbers.

First tower-type lower bounds on generalized Ramsey numbers r_k(t;q,p).

Development of a new variant of the stepping-up lemma for these bounds.

Abstract

Resolving a problem of Conlon, Fox, and R\"{o}dl, we construct a family of hypergraphs with arbitrarily large tower height separation between their $2$-colour and $q$-colour Ramsey numbers. The main lemma underlying this construction is a new variant of the Erd\H{o}s--Hajnal stepping-up lemma for a generalized Ramsey number $r_k(t;q,p)$, which we define as the smallest integer $n$ such that every $q$-colouring of the $k$-sets on $n$ vertices contains a set of $t$ vertices spanning fewer than $p$ colours. Our results provide the first tower-type lower bounds on these numbers.