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arXiv · 1610.06359

Discrepancy and large dense monochromatic subsets

Ross Kang·Viresh Patel·Guus Regts
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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.

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Ross Kang, Viresh Patel, Guus Regts. 2016-10-20. Discrepancy and large dense monochromatic subsets. https://arxiv.org/abs/1610.06359

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