What was already known
Shader proved the right-triangle case; later work treated further triangle families and restricted colorings. The new proof adapts OpenAI’s wild-character rigidity and measurable-transfer machinery, extending its carrier field to handle arbitrary real side-length ratios.
AI involvement
Plasma AI reports unattended alternating research runs by GPT-6 Astra and Claude Opus 5.5 that found the argument. Human direction covered the target and formalization/publication setup; no human mathematical proof input is recorded. The original research contribution, rather than the later formalization management, supports AI-autonomous.
Importance
Resolves the specified longstanding Erdős question, subject to the stated scope. The earlier theorems and later human work retain their credit.
Limitations
Copies allow rotations, reflections and relabelling, but not rescaling. This is the triangle-by-triangle statement: it does not show that one fixed color contains every triangle. That stronger Erdős–Graham formulation remains open. The prose and formalization were produced by AI; the human lead checked three source-matched statements, not the two added variants or the complete proof.
Technical verification
Palomar reports exact selected-statement comparison and kernel checking with Lean, NanoDa and con-ron on 19 October. No local proof rebuild was performed.
Formal checking
Palomar reports exact selected-statement comparison and kernel checking with Lean, NanoDa and con-ron on 19 October. No local proof rebuild was performed.
- Checker
- Palomar: Lean, NanoDa and con-ron
- Scope
- Five selected theorems, including arbitrary two-colorings, at most one missing triangle, the non-equilateral case and three distinct collinear points; Mathlib definitions and only standard Lean axioms.
- Basis
- Reported by the linked checker
- Check date
- 2026-10-19
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. The available blueprint and proof reviews are AI-produced. Human statement correspondence checks do not document a human explanation of the central argument. Few is the evidence default.
Correction and publication history
Added this sourced contribution with its stated scope, AI attribution, checking evidence and research limits.
Read this published revision