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.
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #20 as open (status updated 31 August 2025).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #20 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12). How it works.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/sunflower-conjecture)
- Claims
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- Verified
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- Disputed
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- Refuted
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- On the literature board
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
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.