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Proof of the Erdős-Szemerédi sunflower conjecture
Abstract
In the 1970s, Erdős and Szemerédi conjectured that for any n ≥ k−1≥ 2, any collection of subsets of [n] not containing a k-sunflower has size at most ckn for some ck<2. In the k=3 case, this follows from the work of Croot, Lev, and Pach and Ellenberg and Gijswijt on the cap-set problem, through a reduction due to Alon, Shpilka, and Umans. We prove the conjecture for every k ≥ 3.
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BibTeX
@misc{bucic2026proof,
title = {{Proof of the Erdős-Szemerédi sunflower conjecture}},
author = {Matija Bucić and Fankang He and Hyunwoo Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26686},
eprint = {2610.26686},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26686},
}
RIS
TY - UNPB
TI - Proof of the Erdős-Szemerédi sunflower conjecture
AU - Matija Bucić
AU - Fankang He
AU - Hyunwoo Lee
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26686
UR - https://arxiv.org/abs/2610.26686
AB - In the 1970s, Erdős and Szemerédi conjectured that for any n ≥ k−1≥ 2, any collection of subsets of [n] not containing a k-sunflower has size at most c_k^n for some c_k<2. In the k=3 case, this follows from the work of Croot, Lev, and Pach and Ellenberg and Gijswijt on the cap-set problem, through a reduction due to Alon, Shpilka, and Umans. We prove the conjecture for every k ≥ 3.
ER -