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Statement

A $w$-set system is a set system $\mathcal F$ on a finite set $X$ whose members all have at most $w$ elements (p. 1); sets $S_1,\ldots,S_r$ form an $r$-sunflower if $S_i\cap S_j=S_1\cap\cdots\cap S_r$ for all $i\ne j$ (Definition 1.1, p. 1). Theorem 1.4 (Main theorem, sunflowers), p. 2: "Let $r\geq 3$. For some constant $C$, any $w$-set system $\mathcal F$ of size $|\mathcal F|\geq(Cr^3\log w\log\log w)^w$ contains an $r$-sunflower."

The paper assumes $\log\log w>0$ throughout and, to handle $w=2$, interprets $\log$ as the logarithm in base $1.9$ (p. 2). It places the theorem against the Erdős–Rado lemma (Lemma 1.2, $|\mathcal F|\ge w!(r-1)^w$ suffices) and the sunflower conjecture (Conjecture 1.3, $c(r)^w$ suffices), replacing the bounds $w^{w(1+o(1))}$ by $(\log w)^{w(1+o(1))}$. Page 12 records later improvements by others: Rao's $(Cr\log(wr))^w$ and the observation of Bell, Chueluecha and Warnke that a small modification gives $(Cr\log w)^w$.

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 1.4 on p. 2 and the remarks on p. 12, read on the page images. Published in Ann. of Math. (2) 194 (2021), no. 3, DOI 10.4007/annals.2021.194.3.5 (Crossref record read; the arXiv listing says v3 "took into account comments from the Annals of Mathematics"); an extended abstract appeared in the proceedings of STOC 2020. The journal text was not compared.

Read depth. Claims checked: Definition 1.1, Lemma 1.2, Conjecture 1.3 and Theorem 1.4 were read clause by clause on the page images of pp. 1--2. The proof was not read.

Proof pointer

Theorem 1.4 follows from Theorem 1.9 (any $w$-uniform family of size at least $(C\alpha^{-2}(\log w\log\log w+(\log(1/\beta))^2))^w$ contains an $(\alpha,\beta)$-robust sunflower) with $\alpha=\beta=1/r$ and Lemma 1.8 (a $(1/r,1/r)$-robust sunflower contains an $r$-sunflower), pp. 3--4; the engine is Theorem 2.5, a spreadness bound proved by repeated random sampling and a set-shrinking argument (Section 2), per the paper's introduction.

Dependencies

Lemma 1.8 (quoted from Lovett, Solomon and Zhang) and the paper's own Theorem 1.9; no external premise beyond these.

Bears on

• Problem 20: with $w=n$ and $r=k\ge3$, the theorem bounds the problem's $f(n,k)$ by $\lceil(Ck^3\log n\log\log n)^n\rceil$, that is $(\log n)^{n(1+o(1))}$ for fixed $k$, under the paper's conventions on $\log$; this is not the $c_k^n$ bound the problem asks for.
• Problem 535: the input of the site's derivation $f_r(N)\le N^{C_r\log\log\log N/\log\log N}$, obtained by inserting this theorem, or its later sharpenings recorded on p. 12, in place of the Erdős–Rado bound in the decomposition Erdős sketches in 1964; the corpus has not checked that derivation, and the paper itself does not discuss the equal-gcd problem.