Bounds for Erdős–Szemerédi sunflower-free families

How large can a family of vectors in [D]^n be without a k-sunflower, in which entries in each coordinate are either all equal or pairwise distinct? For k≥3 this vector problem yields bounds for sunflower-free set families.

Why it matters and what counts as progress

Why it matters: An exponential saving for vector sunflowers supplies the missing quantitative ingredient for a longstanding set-family conjecture.

What counts as progress: Prove an exponential saving over the trivial vector bound, sharpen its dependence on k and D or strengthen the resulting set-family bound.

How the problem developed

1978 · Conjecture [1]
Paul Erdős and Endre Szemerédi Posed the sunflower-free family problem and conjectured a substantially stronger upper bound than the elementary counting estimates then available.
2017 · Theorem [2]
Naslund and Sawin Proved the three-petal weak sunflower case using the cap-set method; this did not settle all numbers of petals.
19 Oct 2026 · Reported theorem [3]
Bucić, He, and Lee Posted a proof of the Erdős–Szemerédi sunflower conjecture for the full stated range of petal counts.

AI contribution

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.

AI involvementAI-centered

ImportanceMajor

Technical verificationFormal (Lean + human)

Human understandingFull