A note on diameter-Ramsey sets. (June 2018) - Corsten, Jan; Frankl, Nóra
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.