Triangles with a maximum angle of 135 degrees mark a sharp geometric boundary where the rules of existence in colored maps suddenly flip.
April 29, 2026
Original Paper
Diameter-Ramsey triangles below the 135°
arXiv · 2604.22090
The Takeaway
Ramsey theory proves that structure will eventually emerge from any sufficiently large amount of chaos. A geometric tipping point at 135 degrees determines if a triangle is guaranteed to exist within a colored set of points. Any triangle with an angle smaller than this threshold must appear, while those with larger angles can stay hidden. The existence of such a sharp and specific geometric boundary was entirely unexpected. It provides a new classification for how shapes are forced to organize themselves in space. This finding helps identify the invisible constraints that govern the formation of patterns in complex data.
From the 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 Rödl 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 triangl