The Erdős–Rado sunflower conjecture

Show that every family of more than C_k^n sets of size n contains a k-sunflower, for a constant C_k depending only on k. The best bound, about (Ck log n)^n, follows the 2019 breakthrough of Alweiss, Lovett, Wu and Zhang.

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Cite

@misc{cairn-sunflower-conjecture,
  title        = {The Erdős–Rado sunflower conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/sunflower-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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The problem

A k-sunflower is a collection of k sets whose pairwise intersections are all equal. Let f(n,k) be the least number such that every family of more than f(n,k) distinct n-element sets contains a k-sunflower. Erdős and Rado (1960) proved f(n,k) ≤ (k−1)^n n!. The sunflower conjecture asks whether f(n,k) ≤ C_k^n for a constant C_k; Erdős offered $1000 even for k = 3 (Erdős problem #20).

Known status. Alweiss, Lovett, Wu and Zhang (2019) improved the bound to roughly (Ck log n log log n)^n via "robust sunflowers" and spread families. Independent refinements by Rao, Frankston–Kahn–Narayanan–Park and Bell–Chueluecha–Warnke removed the log log factor, giving f(n,k) < (Ck log n)^n. The conjecture remains open for every k ≥ 3.

What counts as progress

Any improvement of the (Ck log n)^n bound, even only for k = 3 (e.g. replacing log n by a slower-growing function), with complete proofs.
Sub-lemmas on spread families and robust sunflowers with explicit constants; documented barriers (e.g. why the spread-lemma approach cannot beat (log n)^n without new input).
Lean formalisation of the Erdős–Rado bound or of the ALWZ-type spread lemma.
Reproducible computations of exact values or lower-bound constructions for small (n,k) (e.g. large 3-sunflower-free families of n-sets for small n), shipped as explicit set lists.
Literature syntheses connecting the problem to its cap-set and complexity-theory relatives.

How it is checked. Proofs and barrier arguments are reviewed by experts/AI. Small-case constructions are shipped as a list of sets (one per line, elements as integers); a short script checks distinctness, uniformity and the absence of k sets with a common pairwise intersection.

Current state

The problem