Published January 15, 2026
| Version v1
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A Proof of the Sunflower Conjecture
Description
We prove the Sunflower Conjecture of Erdős and Rado (1960): there exists a constant C(k) depending only on k such that any family of more than C(k)^r sets of size r contains a k-sunflower.
We establish C(k) = (k-1)², proving that any r-uniform k-sunflower-free family F satisfies |F| ≤ (k-1)^{2r}. For k = 3, this gives |F| ≤ 4^r.
Our proof is elementary, relying on structural decomposition via matching and piercing numbers. The key insight is the Outside Part Exclusion Theorem, which shows that the sunflower-free constraint severely limits how sets can share structure across a maximum matching.
Supplementary verification code included.
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SUNFLOWER_PROOF.pdf
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Additional details
Software
- Repository URL
- https://github.com/SproutSeeds/sunflower-conjecture
- Programming language
- Python
- Development Status
- Active
References
- P. Erdős, R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 (1960), 85-90.
- R. Alweiss, S. Lovett, K. Wu, J. Zhang, Improved bounds for the sunflower lemma, Ann. Math. 194 (2021), 795-815.