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Statement

Definitions (pp. 2--3). For a finite set $X$, write $\mathcal U(X,p)$ for the random subset $R\subseteq X$ containing each element independently with probability $p$. For $0<\alpha,\beta<1$, a set system $\mathcal F$ on $X$ is $(\alpha,\beta)$-satisfying (Definition 1.5, p. 2) if, with probability greater than $1-\beta$ over $R\sim\mathcal U(X,\alpha)$, some member of $\mathcal F$ is contained in $R$. The link of $\mathcal F$ at $T\subseteq X$ is $\mathcal F_T=\{S\setminus T: S\in\mathcal F,\ T\subseteq S\}$ (p. 3). With $K$ the intersection of all members of $\mathcal F$, the system $\mathcal F$ is an $(\alpha,\beta)$-robust sunflower with kernel $K$ (Definition 1.7, p. 3) if $K\notin\mathcal F$ and the link $\mathcal F_K$ is $(\alpha,\beta)$-satisfying.

Theorem 1.9 (Main theorem, robust sunflowers), p. 4: "Let $0<\alpha,\beta<1$. For some constant $C$, any $w$-uniform set system $\mathcal F$ of size $|\mathcal F|\geq\left(\frac{C}{\alpha^2}\cdot\left(\log w\log\log w +\left(\log\frac1\beta\right)^2\right)\right)^w$ contains an $(\alpha,\beta)$-robust sunflower."

The paper's conventions of p. 2 apply: $\log\log w>0$ is assumed throughout, and $\log$ is read in base $1.9$ to handle $w=2$. The paper notes (p. 4) that for fixed $\alpha,\beta$ the bound $(\log w)^{w(1+o(1))}$ cannot be improved beyond $(\log w)^{w(1-o(1))}$, by Lemma 3.1, and (p. 3) that the theorem verifies a conjecture of Lovett, Solomon and Zhang (its [16]) and answers a question of Rossman (its [21]). On p. 12 it records Rao's later improvement of this bound to $((C/\alpha)\log(w/\beta))^w$.

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 1.9 on p. 4, Definitions 1.5 and 1.7 on pp. 2--3, the remark on p. 12; published in Ann. of Math. (2) 194 (2021), no. 3. The journal text was not compared.

Read depth. Claims checked: Definitions 1.5 and 1.7 and Theorem 1.9 were read clause by clause on the page images of pp. 2--4. The proof was read for structure only.

Proof pointer

Section 2 (pp. 5--11). Lemma 2.4 (p. 6), which the paper repeats from its [16], shows that if the weight profile $(1;\kappa^{-1},\ldots,\kappa^{-w})$ is $(\alpha,\beta)$-satisfying for a nondecreasing $\kappa=\kappa(w)>1$, then every $w$-uniform system of size greater than $\kappa^w$ contains an $(\alpha,\beta)$-robust sunflower: either the system is spread, or a maximal over-full link is passed to. Theorem 1.9 then follows from the bound on the least such $\kappa$ in Theorem 2.5.

Dependencies

Lemma 2.4 (from Lovett, Solomon and Zhang, reproved in the paper) and Theorem 2.5, which supplies the spreadness bound.

Bears on

• Problem 20: through Lemma 1.8 (p. 3, from Lovett, Solomon and Zhang: a $(1/r,1/r)$-robust sunflower contains an $r$-sunflower), the case $\alpha=\beta=1/r$ gives Theorem 1.4, the paper's bound on the sunflower function; it does not give the $c_k^n$ bound the problem asks for.