Monochromatic Triangles

Definitions and mathematical statement

Let $T=\{a,b,c\}\subset\mathbb R^2$ be the three distinct, noncollinear vertices of a prescribed triangle. A three-coloring of the plane is a map $\chi:\mathbb R^2\to\{1,2,3\}$, with no measurability or continuity assumption. A congruent copy of $T$ is $g(T)$ for a Euclidean isometry $g$; translations, rotations, and reflections are allowed, but scaling is not.

For every triangle $T$, does there exist such a coloring with \[ \{\chi(g(a)),\chi(g(b)),\chi(g(c))\} \text{ having at least two elements} \quad\text{for every isometry }g\,? \] Equivalently, can one partition the plane into three color classes so that no class contains a congruent copy of this particular triangle?

The coloring may depend on the triangle, including its side lengths. The problem does not ask for one coloring that avoids all triangle shapes simultaneously. Only the colors of the three vertices matter: colors along the edges or in the interior of their convex hull play no role. Using at most three colors is permitted. Replacing congruence by similarity, imposing regularity on the color classes, or requiring a single coloring for every $T$ would produce stronger and different questions.

Short English statement

For each prescribed triangle, can the plane be colored with three colors so that no congruent copy has all three vertices the same color?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 5500058 (AMR-054-0058), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problems Project - Computational Geometry, Problem 58. Original source indexed by AMR-054-0058. https://topp.openproblem.net/p58.
• S3 The Open Problems Project, original numbered problem and status notes. https://topp.openproblem.net/p58.