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Statement

Theorem 4.1 (p. 13): "For any $r\geq3$ there exists $c=c(r)$ such that the following holds. Let $\mathcal F$ be a set system on $X$, with $|X|=n$ and $|\mathcal F|\geq2^{n(1-c/\log n)}$. Then $\mathcal F$ contains an $r$-sunflower."

Here an $r$-sunflower is $r$ sets whose pairwise intersections all equal their common intersection (Definition 1.1, p. 1); the sets of $\mathcal F$ may have any sizes. The paper places the theorem (p. 13) against the Erdős–Szemerédi sunflower conjecture, that the bound can be improved to $2^{n(1-\varepsilon)}$ for some $\varepsilon=\varepsilon(r)$, which it says Naslund proved for $r=3$ by algebraic techniques (its [18], Naslund and Sawin).

Source. R. Alweiss, S. Lovett, K. Wu and J. Zhang, Improved bounds for the sunflower lemma, arXiv:1908.08483v3 (31 August 2021, 19 pages; the copy read), Theorem 4.1 on p. 13; published in Ann. of Math. (2) 194 (2021), no. 3. The journal text was not compared.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 13. The paper gives no proof of its own to check.

Proof pointer

The paper gives none beyond two sentences (p. 13): Erdős and Szemerédi (P. Erdős and E. Szemerédi, Combinatorial properties of systems of sets, J. Combin. Theory Ser. A 24 (1978), 308--313, the paper's [8]) show that the Erdős–Rado sunflower conjecture implies their conjecture, and inserting the new bounds into their argument gives Theorem 4.1. The input is the paper's sunflower bound, Theorem 1.4, or its refinements; the paper does not say which form is used.

Dependencies

The paper's new sunflower bounds, of which Theorem 1.4 is the main one, and the Erdős–Szemerédi reduction of 1978, which the paper cites and does not reproduce.

Bears on

• Problem 857: with $k=r\ge3$, the statement gives $m(n,k)\le\lceil2^{n(1-c(k)/\log n)}\rceil$ for the problem's least $m$ forcing a $k$-sunflower among subsets of $\{1,\ldots,n\}$, an upper bound and not the estimate or asymptotic formula the problem asks for; the paper derives it only by citing the Erdős–Szemerédi argument.