Hedgehogs are not colour blind

TL;DR

This paper presents a family of hypergraphs demonstrating that Ramsey numbers can grow polynomially with vertices for fewer colours but exponentially for more, highlighting a strong colour dependence.

Contribution

It introduces the first known class of hypergraphs with Ramsey numbers that significantly depend on the number of colours used.

Findings

2-colour Ramsey numbers grow polynomially

4-colour Ramsey numbers grow exponentially

Shows strong colour dependence in hypergraph Ramsey numbers

Abstract

We exhibit a family of $3$-uniform hypergraphs with the property that their $2$-colour Ramsey numbers grow polynomially in the number of vertices, while their $4$-colour Ramsey numbers grow exponentially. This is the first example of a class of hypergraphs whose Ramsey numbers show a strong dependence on the number of colours.