@inbook{99ef0846559248eb9c4753e01eeb3b94,
title = "On Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs",
abstract = "For n≥ s> r≥ 1 and k≥ 2, write n→(s)kr if every k-colouring of all r-subsets of an n-element set has a monochromatic subset of size s. Improving upon previous results by Axenovich et al. (Discrete Mathematics, 2014) and Erd{\H o}s et al. (Combinatorial set theory, 1984) we show that ifr≥3andn↛(s)krthen2n↛(s+1)k+3r+1. This yields an improvement for some of the known lower bounds on multicolour hypergraph Ramsey numbers. Given a hypergraph H= (V, E), we consider the Ramsey-like problem of colouring all r-subsets of V such that no hyperedge of size ≥ r+ 1 is monochromatic. We give upper and lower bounds on the number of colours necessary in terms of the chromatic number χ(H). We show that this number is O(log(r-1)(rχ(H) ) + r).",
keywords = "Hypergraph colouring, Ramsey numbers, Stepping-up lemma",
author = "Bruno Jartoux and Chaya Keller and Shakhar Smorodinsky and Yelena Yuditsky",
note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.",
year = "2021",
month = jan,
day = "1",
doi = "https://doi.org/10.1007/978-3-030-83823-2_81",
language = "American English",
series = "Trends in Mathematics",
publisher = "Springer Science and Business Media Deutschland GmbH",
pages = "503--508",
booktitle = "Trends in Mathematics",
address = "Germany",
}