Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3

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Key facts

• Paper: ‘Proof of the Erdős-Szemerédi sunflower conjecture’, Matija Bucić, Fankang He (Faculty of Mathematics, University of Vienna), Hyunwoo Lee (KAIST and IBS ECOPRO), arXiv 2610.26686 [math.CO], submitted 19 Oct 2026
• 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 c_k^n for some c_k < 2.’ The paper traces the question back to 1970
• Theorem 1: any subset of 2^[n] with no sunflower of k sets has size at most 2^((1 − 1/(16k³))n). Lower bound (Deuber, Erdős, Gunderson, Kostochka, Meyer): k-sunflower-free families of size 2^(n − O(n log log k / log k)), so the result is ‘not far from best possible for large k’
• Earlier: Erdős–Szemerédi 2^(n − Ω(√n)); Alweiss–Lovett–Wu–Zhang (2019) and Rao give 2^(n − Ω(n/log n)). k = 3 followed from the 2016 cap-set bounds (Croot–Lev–Pach; Ellenberg–Gijswijt) through Alon–Shpilka–Umans; Naslund and Sawin gave base 3/2^(2/3) ≈ 1.89 using Tao’s slice rank
• Method: Theorem 2 bounds the number V_k(D, n) of vectors in [D]^n without a k-sunflower by the polynomial method (dimension counting with low-degree factors and a Chernoff bound); a reduction (Section 4) transfers the vector bound to set families with D = 16k²
• AI statement, verbatim: ‘The proof of Theorem 2 was found by ChatGPT 6 Astra Pro, following a short discussion, which started by asking it to generalize the k = 3 case argument in [45], which we needed for our original target of proving the induced variant of the Erdős–Sauer problem. The original argument it produced used the slice-rank terminology due to Tao (as used in [45]); we rewrote it in the polynomial method language … The authors have verified all the results in the paper and take full responsibility for the content. The proof has been verified in Lean 4.35; the verification will be made available on the Palomar repository.’ ([45] is Naslund–Sawin)
• Concurrent work cited in the paper: another team proves that the set-sunflower relation on the Boolean cube has maximal slice rank, and ‘They discuss extensive use of AI, including multi-agent setups’. The authors say that obstruction concerns set sunflowers, while their tester is for vector sunflowers, so it does not apply
• Consequences listed in the paper: the vector bound extends the Alon–Shpilka–Umans question (connected to fast matrix multiplication) and gives results in arithmetic property testing and on equal subset sums