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Statement

Setting (p. 2). For a finite set $X$, $X_\delta$ is the random subset of $X$ containing each element independently with probability $\delta$; $r$-spread is as in Lemma 2.

Theorem 3 (Main technical estimate of Rao and Tao, p. 2). There is a constant $B\geq1$ such that for every integer $k\geq2$, all reals $0<\delta,\epsilon\leq1/2$ and $r\geq B\delta^{-1}\log(k/\epsilon)$, and every family $\mathcal S$ of $k$-element subsets of a finite set $X$: if $\mathcal S$ is $r$-spread and $|\mathcal S|\geq r^k$, then $\mathbb P(\exists S\in\mathcal S:S\subseteq X_\delta)>1-\epsilon$.

The theorem is not the paper's own: the paper attributes it to Rao (Discrete Analysis 2020) and Tao (blog post, 2020), and its appendix records how it follows from their proofs. Lemma 4 shows the spread hypothesis is essentially best possible.

Proof pointer

Appendix, p. 3. The paper says the theorem follows from Tao's proof of his Proposition 5, and gives a short derivation from Rao's proof of his Lemma 4: Rao's argument controls a uniformly random subset of size $\lceil\delta|X|/2\rceil$, and a Chernoff bound transfers this to $X_\delta$, using that $|\mathcal S|\geq r^k$ forces $|X|\geq r$. The resulting constant is $B=\max\{2\alpha,16\}$ with $\alpha$ Rao's constant.

Read depth

Claims checked: Theorem 3 was read clause by clause on p. 2 and the appendix derivation on p. 3 followed; Rao's and Tao's underlying arguments were not read here. Nothing here is independently reviewed.

Dependencies

External: Rao, Coding for sunflowers, Discrete Analysis 2020 (proof of Lemma 4, card rao_2020_coding_sunflowers), and Tao's 2020 blog post; a Chernoff bound from Janson, Łuczak and Ruciński.

Source. T. Bell, S. Chueluecha and L. Warnke, Note on sunflowers, Discrete Math. 344 (2021), no. 7, 112367, doi:10.1016/j.disc.2021.112367; the edition read, arXiv:2009.09327v2, is named on the source card, and the labels and pages here are its.

Bears on

• Problem 20: the input of Lemma 2 and so of Theorem 1; it decides nothing about the problem by itself.