Diameter-Ramsey triangles below the $135^\circ$

Abstract

A finite Euclidean set is diameter-Ramsey if, for every number of colors, some finite same-diameter witness has the property that every coloring of the witness contains a monochromatic congruent copy of the set. Frankl, Pach, Reiher and R\"odl asked whether any obtuse triangle is diameter-Ramsey. We prove the stronger statement that every non-degenerate triangle whose largest angle is strictly smaller than $135^\circ$ is diameter-Ramsey. Together with the theorem of Corsten and Frankl that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, this gives the sharp classification for the two open angular ranges on either side of $135^\circ$. The proof uses a weighted $k$-subset configuration with non-negative coefficients; a finite binary-tree construction realizes the required two prescribed overlaps, and the ordinary hypergraph Ramsey theorem then forces a monochromatic copy of the triangle.