Tower Gaps in Multicolour Ramsey Numbers
Quentin Dubroff, Ant\'onio Gir\~ao, Eoin Hurley, and Corrine Yap

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 -colour and -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 , which we define as the smallest integer such that every -colouring of the -sets on vertices contains a set of vertices spanning fewer than colours. Our results provide the first tower-type lower bounds on these numbers.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
