TY - CHAP

T1 - On Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs

AU - Jartoux, Bruno

AU - Keller, Chaya

AU - Smorodinsky, Shakhar

AU - Yuditsky, Yelena

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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 Erdős 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).

AB - 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 Erdős 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).

KW - Hypergraph colouring

KW - Ramsey numbers

KW - Stepping-up lemma

UR - http://www.scopus.com/inward/record.url?scp=85114105605&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-83823-2_81

DO - 10.1007/978-3-030-83823-2_81

M3 - Chapter

AN - SCOPUS:85114105605

T3 - Trends in Mathematics

SP - 503

EP - 508

BT - Trends in Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -