Tower Gaps in Multicolour Ramsey Numbers

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 Erdős--Hajnal stepping-up lemma f...

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Bibliographic Details
Published inarXiv.org
Main Authors Dubroff, Quentin, Girão, António, Hurley, Eoin, Yap, Corrine
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 01.09.2023
Subjects
Apexes
Color
Coloring
Graph theory
Lower bounds
Mathematics - Combinatorics
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Summary: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 Erdős--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.
ISSN:2331-8422
DOI:10.48550/arxiv.2202.14032