A note on diameter-Ramsey sets. (June 2018)

• Record Type: Journal Article
• Authors: Corsten, Jan; Frankl, Nóra
• Is Part Of: European journal of combinatorics. Volume 71(2018)
• Page Start: 51
• Page End: 54
• Publication Date: 2018-06
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2018.02.036
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: A finite set A ⊂ R d is called diameter-Ramsey if for every r ∈ N, there exists some n ∈ N and a finite set B ⊂ R n with diam ( A ) = diam ( B ) such that whenever B is coloured with r colours, there is a monochromatic set A ′ ⊂ B which is congruent to A . We prove that sets of diameter 1 with circumradius larger than 1 ∕ 2 are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than 135° are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher and Rödl. Furthermore, we deduce that there are simplices which are almost regular but not diameter-Ramsey.