Erdős Problem #173
In any 2-colouring of ℝ^2, for all but at most one triangle T (up to congruence), there is a monochromatic congruent copy of T. The colouring is arbitrary. No measurability is assumed. ≅ compares vertices with the same label, so the conclusion allows a relabelling σ.
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #173 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_173as open.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #173 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_173` | research open | Formal Conjectures |
| Formal Conjectures: research statements in the file | 1 open, 2 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.
From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.
Cite
@misc{cairn-erdos-173,
title = {Erdős Problem #173},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-173}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-173)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.
- Get a task — one click
- Your chatbot works on it — 5–10 minutes, in its own tab
- Paste its answer here — check it and publish it under your name
What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
In any -colouring of , for all but at most one triangle (up to congruence), there is a monochromatic congruent copy of .
The colouring is arbitrary. No measurability is assumed. ≅ compares vertices with the same label, so the conclusion allows a relabelling σ.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«173».
theorem erdos_173 : ∀ (c : ℝ² → Fin 2) (T₁ T₂ : Triangle ℝ ℝ²),
¬ HasMonochromaticCopy c T₁ → ¬ HasMonochromaticCopy c T₂ →
∃ σ : Equiv.Perm (Fin 3), T₁.points ∘ σ ≅ T₂.pointsWhat counts as progress
- A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
- Partial results: special cases, weaker bounds, reductions — as verified claims.
- Computations and numerical evidence with published code (reproducible).
- Literature: the problem may have been solved or partly solved already — check erdosproblems.com/173. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
- erdosproblems.com/173
- [Sh76] Shader, L. E., All right triangles are Ramsey in !. J. Combin. Theory Ser. A (1976), 385-389.
Source and licence
Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.