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Statement

Lemma 3.1 (p. 11): "There exists a $w$-set system of size $((\log w)/8)^{w-\sqrt w}=(\log w)^{w(1-o(1))}$ which does not contain a $(1/2,1/2)$-robust sunflower."

Section 3 assumes, just before the lemma (p. 11), that $w$ is sufficiently large, and fixes $\alpha=\beta=1/2$ for concreteness, saying that the construction can be easily modified for any other constant values of $\alpha,\beta$. Robust sunflowers are defined on the page of Theorem 1.9, which the lemma shows to be tight up to the $o(1)$ in the exponent for $\alpha=\beta=1/2$. The lemma concerns robust sunflowers only; it says nothing about ordinary sunflowers.

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), Lemma 3.1 on p. 11, proof on pp. 11--12; published in Ann. of Math. (2) 194 (2021), no. 3. The journal text was not compared.

Read depth. Claims checked: the statement and the standing assumption of Section 3 were read clause by clause on the page image of p. 11. The proof was read for structure only.

Proof pointer

Pages 11--12: take $w$ disjoint blocks of size $\log(w/c)$, with $c=1/\varepsilon$, and the system of all transversals, which is not $(1/2,1/2)$-satisfying (Claim 3.2); a greedy subsystem with pairwise intersections at most $(1-\varepsilon)w$ forces every robust sunflower to have a kernel too small for its link to be satisfying (Claim 3.3), and $\varepsilon=1/\sqrt w$ gives the stated size.

Dependencies

None outside the paper's definitions.

Bears on

No Erdős problem directly: it limits the robust-sunflower method of Theorem 1.9 behind the paper's bound for Problem 20, not the sunflower function itself.