Discrepancy and large dense monochromatic subsets

TL;DR

This paper explores generalized Ramsey numbers focusing on monochromatic subgraphs with high degree in multi-colour and hypergraph settings, revealing phase transitions and introducing a new discrepancy measure.

Contribution

It extends the study of quasi-Ramsey numbers to multiple colours and hypergraphs, highlighting phase transitions and proposing a novel density-biased discrepancy concept.

Findings

Quasi-Ramsey numbers exhibit phase transitions across various settings.

A new density-biased hypergraph discrepancy measure is introduced.

Results apply to both high minimum degree and average degree monochromatic subgraphs.

Abstract

Erd\H{o}s and Pach (1983) introduced the natural degree-based generalisations of Ramsey numbers, where instead of seeking large monochromatic cliques in a $2$-edge coloured complete graph, we seek monochromatic subgraphs of high minimum or average degree. Here we expand the study of these so-called quasi-Ramsey numbers in a few ways, in particular, to multiple colours and to uniform hypergraphs. Quasi-Ramsey numbers are known to exhibit a certain unique phase transition and we show that this is also the case across the settings we consider. Our results depend on a density-biased notion of hypergraph discrepancy optimised over sets of bounded size, which may be of independent interest.