Chvátal's 1972 conjecture proved (Chang–Liu–Liu, ChatGPT-assisted), then a GPT-6 Astra 'proof from The Book' and a Codex-built Lean formalization
On Sept 16, 2026 Fan Chang, Hong Liu and Miao Liu posted a proof of Chvátal's conjecture (1972): every hereditary family of sets has a largest intersecting subfamily that is a star. It went through a sharp correlation inequality for increasing Boolean functions. ChatGPT tested candidate inequalities and proved a special case that pointed the authors to the general theorem. Within days the proof was formalised in Lean with GPT-6 via Codex. On Sept 23 Ellis, Filmus and Friedgut posted a 7-page spectral "proof from The Book" that GPT-6 Astra produced after reading the Chang–Liu–Liu manuscript.
Key facts
- arXiv 2609.19123 (Sept 16, 2026), Chang, Liu & Liu: proves Chvátal's conjecture and the Friedgut–Kahn–Kalai–Keller correlation form Cov(f,g) ≥ ¼ min_i Inf_i[f] for antipodal g
- AI disclosure (Chang–Liu–Liu): 'We used ChatGPT to test candidate inequalities… ChatGPT proved the special case of the inequality in Theorem 1.2 when f is a symmetric threshold function, inspiring the authors to pursue and ultimately prove the stronger inequality'; all proofs written and checked by the authors
- Lean: boonsuan/chvatal formalizes the Chang–Liu–Liu proof, 'generated with GPT-6 through Codex'; registered with Palomar as PALOMAR-2026-09-17-000004; no sorry or extra axioms in the library; also imported into Lean Pool (PR #443)
- arXiv 2609.28404 (Sept 23), Ellis, Filmus & Friedgut, 'Chvátal's conjecture: a proof from The Book': a short direct spectral proof plus a projection-packing-number strengthening
- Their AI chronology: weeks of failed attempts with ChatGPT-6 Astra and Claude Fable 5.1 on their own conjectures (H, I); after they fed in the Chang–Liu–Liu paper, Astra announced a proof that the projection packing number is tight for every downset, then 'stripped [it] down to the spectral bare bones'; ChatGPT used only for proofreading of the final paper
Science result
- Field
- mathematics / extremal combinatorics / Boolean functions
- Problem
- Chvátal's conjecture on intersecting subfamilies of downsets (open since 1972)
- Result
- Proof of Chvátal's conjecture via a sharp correlation inequality, followed by an AI-found short spectral proof and a Lean formalization.
- AI system
- ChatGPT, GPT-6 Astra, Codex
- Human role
- AI-assisted, human-led original proof (ChatGPT tested inequalities and proved a guiding special case); the short spectral proof was found by GPT-6 Astra under human direction
- Verification
- Formal proof in Lean (Codex-generated formalization of the Chang–Liu–Liu proof); preprints not yet refereed
- Status
- confirmed
- Why surprising
- A 54-year-old conjecture fell, and within a week a model had shortened the proof to a few pages and another had machine-checked it.
What happened
Chvátal's conjecture, posed in 1972, is one of the best-known open problems in extremal set theory. Chang, Liu and Liu (IBS) proved it through a sharp correlation inequality for increasing Boolean functions, and they credit ChatGPT with proving a special case that showed them the way. A Lean formalization made with GPT-6 in Codex appeared the next day in the Palomar registry. A week later, Ellis, Filmus and Friedgut described how GPT-6 Astra had failed for weeks on their related conjectures. Once given the new paper, it produced a short spectral proof, which they present as "a proof from The Book".
Why it matters
It is a clear example of the late-2026 pattern in mathematics. A human-led breakthrough with model assistance is followed within days by AI simplification and AI formalization. Disclosures are now detailed enough to separate the contribution of each model.
Changelog
- 2026-10-01: created (leads run, 07:40 completion)
Related events
- Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) ★★★★
- Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims ★★★
- Lean Pool: an AI-maintained archive of Lean formalizations grows past 3 million lines ★★
- GPT-6 Astra finds a proof improving the Kővári–Sós–Turán bound: diagonal bipartite Ramsey numbers b(t,t) = O(2^t) (Mubayi) ★★★
Sources (4)
- paperarXiv 2609.19123: A proof of Chvátal's conjecture via a sharp correlation inequality
- paperarXiv 2609.28404: Chvátal's conjecture, a proof from The Book (Ellis, Filmus, Friedgut)
- codeGitHub boonsuan/chvatal: Lean formalization generated with GPT-6 through Codex
- codeLean Pool PR #443: import Chvátal's conjecture and sharp Boolean correlation
id: 2026-09-16-chvatal-conjecture-proved · updated 2026-10-01 · open in the interactive timeline