E173: “Can at most one triangle class lack a monochromatic congruent copy in every plane two-colouring?”

Open

Statement	User	Model	Harness	Time
2)V2Every triangle has a monochromatic congruent copy under either constant two-colouring of the plane.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every two-colouring of the Euclidean plane, any two noncongruent nondegenerate triangle shapes cannot bot…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 Every triangle has a monochromatic congruent copy under either constant two-colouring of the plane.

kernel-checked, filed Tue Aug 25 2026 07:51:30 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Both constant point two-colourings of the Euclidean plane and every labelled triangle.

1) V1 For every two-colouring of the Euclidean plane, any two noncongruent nondegenerate triangle shapes cannot both lack monochromatic congruent copies.

open, filed Tue Aug 25 2026 07:51:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Allowing a vertex permutation is essential: indexwise distance equality would distinguish relabellings of the same triangle. The pairwise proposition is the direct at-most-one formulation on congruence classes.

Scope. All point two-colourings of the Euclidean plane and all pairs of noncongruent nondegenerate triangle classes.

E173:Erdős Problem #173 “Can at most one triangle class lack a monochromatic congruent copy in every plane two-colouring?”

E173: “Can at most one triangle class lack a monochromatic congruent copy in every plane two-colouring?”