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Statement

Definitions (pp. 5--6). A weighted set system $(\mathcal F,\sigma)$ gives the members of $\mathcal F$ nonnegative rational weights, not all zero, with $\sigma(\mathcal F')$ the total weight of $\mathcal F'\subseteq\mathcal F$. A weight profile is a vector $\mathbf s=(s_0;s_1,\ldots,s_k)$ with $s_0\ge s_1\ge\cdots\ge s_k\ge0$ and $s_0>0$, read as $s_i=0$ for $i>k$. For $\mathbf s=(s_0;s_1,\ldots,s_w)$, the system $(\mathcal F,\sigma)$ is $\mathbf s$-spread (Definition 2.1, p. 5) if $\sigma(\mathcal F)\ge s_0$ and $\sigma(\mathcal F_T)\le s_{|T|}$ for every link $\mathcal F_T$ at a nonempty $T$; in particular $\mathcal F$ is then a $w$-set system. A set system is $\mathbf s$-spread if some weight function makes it so (Definition 2.2), and for $0<\alpha,\beta<1$ the profile $\mathbf s$ is $(\alpha,\beta)$-satisfying if every $\mathbf s$-spread set system is $(\alpha,\beta)$-satisfying (Definition 2.3, p. 6; satisfying set systems are defined on the page of Theorem 1.9). For $0<\alpha,\beta<1$ and $w\ge2$, $\kappa(w,\alpha,\beta)$ is the least $\kappa$ such that $(1;\kappa^{-1},\ldots,\kappa^{-w})$ is $(\alpha,\beta)$-satisfying (p. 6).

Theorem 2.5 (p. 6): "$\kappa(w,\alpha,\beta)=O\left(\frac1{\alpha^2}\cdot\left(\log w\log\log w +\left(\log\frac1\beta\right)^2\right)\right)$."

The end of the proof (p. 11) shows that the conclusion holds whenever

$$\kappa=\Omega\Bigl(\max\Bigl\{\Bigl(\frac1\alpha\Bigr)^{1+2/\log\log w} \log w\log\log w,\ \frac1\alpha\Bigl(1+\log\frac1\beta\Bigr)^2, \ \frac1\alpha\Bigl(1+\log\frac1\beta\Bigr)\log\log w\Bigr\}\Bigr),$$

a finer form from which the stated bound follows. On p. 13 the paper quotes Rao's later bound $\kappa(w,\alpha,\beta)\le(C/\alpha)\log(w/\beta)$ for some constant $C$, and uses it, not Theorem 2.5, for its applications in Section 4.

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 2.5 on p. 6, Definitions 2.1--2.3 on pp. 5--6, the end of the proof on p. 11; published in Ann. of Math. (2) 194 (2021), no. 3. The journal text was not compared.

Read depth. Claims checked: Definitions 2.1--2.3, the definition of $\kappa(w,\alpha,\beta)$ and Theorem 2.5 were read clause by clause on the page images of pp. 5--6, and the parameter choice on p. 11. The proof (pp. 6--11) was not checked.

Proof pointer

Section 2 (pp. 6--11): the reduction step, Lemma 2.6 (p. 6), samples a $p$-biased random set $W$ and replaces most sets $S$ by a set $S'\setminus W$ of size at most $w'\le w$, losing little spreadness; it is proved by an encoding argument modelled on Razborov's proof of Håstad's switching lemma (p. 5). Iterating it, at most $(K\log w)/\varepsilon$ times with $\varepsilon=1/\log\log w$ (pp. 10--11), shrinks the sets, and Lemma 2.10, proved by Janson's inequality, finishes; the parameters are chosen on p. 11.

Dependencies

Lemma 2.6 and Lemma 2.10, the latter proved in the paper by Janson's inequality.

Bears on

• Problem 20: indirectly, as the input of Theorem 1.9 and hence of Theorem 1.4.