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Proof of the Erdős-Szemerédi sunflower conjecture

Matija Bucić, Fankang He, Hyunwoo Lee

For integers k≥3 and D,n≥1, a k-sunflower-free family in [D]^n has size at most ((2k)^(2/(k−1)) D^(1−1/(k−1)))^n. Here a vector sunflower has entries all equal or pairwise distinct in every coordinate. The authors’ reduction gives the set-family bound 3(2−1/(16k³))^n and resolves the Erdős–Szemerédi conjecture for every k≥3.

Sources for this record

What was already known

The k=3 case followed from the 2016 cap-set breakthrough and earlier vector-to-set reductions. The general k case remained open. GPT supplied Theorem 2; the authors rewrote its slice-rank proof in polynomial-method language and completed the quantitative reduction to set families.

AI involvement AI-centered

The authors credit ChatGPT 6 Astra Pro with finding the proof of Theorem 2 after a short discussion asking it to generalize Naslund–Sawin’s k=3 argument. Humans posed that direction, rewrote the reasoning and supplied the reduction and applications. The retained AI contribution is the central vector theorem, not sole authorship of every result.

Importance Major

The central vector bound supplies the missing exponential saving for a longstanding extremal-set problem.

Limitations

This is the nonuniform Erdős–Szemerédi sunflower conjecture. The uniform Erdős–Rado sunflower conjecture remains open. Human checking is reported by the authors; the promised Lean artifact is not yet available in the reviewed public sources.

Technical verification Formal (human)

Bucić, He and Lee explicitly report verifying all results, including the model-found vector theorem. This supports complete source-reported human checking. They also report Lean 4.35 verification, but the promised public artifact was not located, so machine statement correspondence is not established here.

Independent checking

Bucić, He and Lee explicitly report verifying all results, including the model-found vector theorem. This supports complete source-reported human checking. They also report Lean 4.35 verification, but the promised public artifact was not located, so machine statement correspondence is not established here.

Checker
Matija Bucić, Fankang He and Hyunwoo Lee (authors)
Scope
Theorem 2: the stated bound for k-sunflower-free vectors for all integers k>=3 and D,n>=1; authors also explicitly checked every result in the paper.
Basis
Reported by the linked checker
Evidence source

Human understanding Full

Bucić, He and Lee rewrite and explain the model’s central interpolation/dimension-counting mechanism in Sections2–3, then explain how their block reduction transfers the vector saving to set families.

Correction and publication history

2026-10-22 · publication

Added the sourced contribution with its stated hypotheses, AI role, prior work and checking limits.

Read this published revision