Monochromatic triangles under any 2-colouring of the plane: at most one exceptional shape? (Erdős #173)
Statement
Prove that in any $2$-colouring of the plane $\mathbb{R}^2$, for all but at most one triangle $T$ (up to congruence) there is a monochromatic congruent copy of $T$ — three points of a single colour forming a triangle congruent to $T$. Here a triangle is any three non-collinear points, and the single possible exception is understood to be an equilateral triangle of one fixed size.
Acceptance. FULLY RESOLVES: a complete proof that every $2$-colouring of $\mathbb{R}^2$ contains a monochromatic congruent copy of $T$ for all but at most one congruence class of triangle $T$. A machine-checkable proof (Lean/Coq) is preferred, otherwise a full written proof with all steps. ADVANCES: (a) prove the statement for a new nontrivial class of triangles beyond Shader's right-angled case (for example all isosceles triangles, or all triangles containing a fixed angle), with proof; (b) prove that at most one exceptional triangle can occur within a restricted family of colourings; (c) exhibit, via a verified finite point configuration, that a specific triangle $T$ forces a monochromatic congruent copy in every $2$-colouring, delivered as a SAT or enumeration certificate. Deliver the proof file or the finite-configuration certificate with solver logs.
Background
Asked by Erdős [Er75f, p.108] and reiterated by Erdős and Graham [ErGr79], [ErGr80], [Er83c]. This is a Euclidean Ramsey question. That at least one exception can be forced is shown by the alternating parallel-strips colouring: colouring $\mathbb{R}^2$ by horizontal strips of a suitable width in two alternating colours leaves one particular equilateral triangle with no monochromatic congruent copy, so 'all but at most one' would be best possible. Toward the conjecture, Shader [Sh76] proved the statement when restricted to a single right-angled triangle: every $2$-colouring of the plane contains a monochromatic congruent copy of any given right triangle. The general claim (all triangles but at most one) remains open. No cash prize is attached. Listed as open on erdosproblems.com/173 (fetched 2026-07-13, status 'open', tagged 'geometry | ramsey theory'). The attacker's tool: proof-shaped combinatorial geometry — compactness and colouring-structure arguments plus explicit configuration constructions; the direct computational purchase is limited, though a SAT or enumeration search over finite planar point sets can certify that a specific triangle forces a monochromatic copy.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #173 (T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.