Monochromatic triangles in two-colourings of the plane
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Must every arbitrary two-coloring of the Euclidean plane contain a monochromatic congruent copy of each non-equilateral three-point set? Three distinct collinear points are included. For equilateral triangles, the target allows at most one missed side length.
Why it matters and what counts as progress
Why it matters: The question asks for geometric patterns forced without measurability or regularity assumptions on the coloring.
What counts as progress: A theorem for every non-equilateral shape and the sharp equilateral exception, or an arbitrary coloring avoiding an additional required shape.
How the problem developed
- 1973 · Conjecture [1]
Paul Erdős, Ronald L. Graham, Peter Montgomery, Bruce L. Rothschild, Joel Spencer and Ernst G. Straus Conjectured that every non-equilateral triangle is forced in every two-colouring of the plane; if an equilateral side length is missed, all other side lengths should occur.
Plasma AI; its proof was found by GPT-6 Astra in Codex and Claude Opus 5.5 in Claude Code The project reports a Lean-checked proof that every non-equilateral triangle, including collinear triples, occurs monochromatically and that at most one equilateral scale can be missed.
AI contribution
Every two-coloring of the plane misses at most one triangle, which is equilateral
In every arbitrary two-coloring of the Euclidean plane, each non-equilateral three-point set, including three distinct collinear points, has a monochromatic congruent copy. At most one equilateral side length can be missed, and an alternating-strip coloring shows that one exception can occur.
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Publication dates and editorial changes.
Every two-coloring of the plane misses at most one triangle, which is equilateral