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Statement

Setting (pp. 3 and 5). A hypergraph $\mathcal H$ on $X$ is a collection of subsets of $X$ with repeats allowed; it is $r$-bounded if each edge has at most $r$ elements, and $\kappa$-spread if $|\mathcal H\cap\langle S\rangle|\le\kappa^{-|S|}|\mathcal H|$ for every $S\subseteq X$, where $\langle S\rangle=\{T\subseteq X:T\supseteq S\}$ and edges are counted with multiplicity. Section 3 fixes a slightly small constant $\gamma$ (it says $\gamma=0.1$ suffices) and a constant $C_0$ large enough for its estimates, and takes $\mathcal H$ an $r$-bounded, $\kappa$-spread hypergraph on a set $X$ of size $n$, with $r,\kappa\ge C_0^2$. It sets $p=C/\kappa$ with $C_0\le C\le\kappa/C_0$ (so $p\le1/C_0$), $r'=(1-\gamma)r$ and $N=\binom{n}{np}$, and fixes a map $\psi:\langle\mathcal H\rangle\to\mathcal H$ with $\psi(Z)\subseteq Z$ for every $Z\in\langle\mathcal H\rangle$, where $\langle\mathcal H\rangle$ is the union of the $\langle S\rangle$ over edges $S$. For $W\subseteq X$ and $S\in\mathcal H$ it puts $\chi(S,W)=\psi(S\cup W)\setminus W$, and calls the pair $(S,W)$ bad if $|\chi(S,W)|>r'$ and good otherwise.

Lemma 3.1 (p. 5). For $\mathcal H$ as above and $W$ chosen uniformly from the $np$-element subsets of $X$, the expected number of edges $S\in\mathcal H$ for which $(S,W)$ is bad is at most $|\mathcal H|C^{-r/3}$.

The paper describes the lemma as an improvement of Lemma 5.7 of the arXiv v1 of Alweiss, Lovett, Wu and Zhang, Improved bounds for the sunflower lemma (p. 5), and says its approach strengthens theirs (p. 4).

Proof pointer

Pp. 6--7. It suffices to bound, for each size $s\in(r',r]$, the number of bad pairs $(S,W)$ with $|S|=s$ by $(\gamma r)^{-1}N|\mathcal H|C^{-r/3}$. The pairs are split by whether $(S,W\cup S)$ is "pathological", meaning that for some $T\subseteq S$ with $t=|T|>r'$ the number of edges of size $s$ containing $T$ and contained in $W\cup S$ exceeds $\sqrt C^{\,r}|\mathcal H|\kappa^{-t}p^{s-t}$, the spread-based estimate times $\sqrt C^{\,r}$. Nonpathological pairs are counted by an encoding in the style of Alweiss, Lovett, Wu and Zhang, now with the sharper count that nonpathology allows; pathological pairs are counted by a Markov bound on the choice of $W\cup S$ outside $S$. The two counts sum to less than the required bound. The new ingredient, by the paper's account (p. 6), is the separate treatment of the pathological part.

Read depth

Claims checked: the Section 3 setting and the lemma were read clause by clause on the page image of p. 5. The proof was read for structure only.

Dependencies

None outside the paper's definitions.

Source. K. Frankston, J. Kahn, B. Narayanan and J. Park, Thresholds versus fractional expectation-thresholds, Ann. of Math. (2) 194 (2021), no. 2, doi:10.4007/annals.2021.194.2.2; the edition read, arXiv:1910.13433v2, is named on the source card, and the labels and pages here are its.

Bears on

• Problem 20: methodological only. The paper calls the lemma an improvement of a lemma from the Alweiss–Lovett–Wu–Zhang sunflower paper, and uses it for thresholds (Theorem 1.1 and Theorem 1.7); the paper states and derives no bound on the sunflower function $f(n,k)$.