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Statement

Setting (p. 1). A family $\mathcal S$ of $k$-element sets is $r$-spread when every non-empty set $T$ is contained in at most $r^{k-|T|}$ members of $\mathcal S$.

Lemma 2 (p. 1). There is a constant $C\geq4$ such that, with $r(p,k)=Cp\log k$, the following holds for all integers $p,k\geq2$: if $\mathcal S$ is a family of at least $r(p,k)^k$ sets of size $k$ and $\mathcal S$ is $r(p,k)$-spread, then $\mathcal S$ contains $p$ disjoint sets.

The paper notes (p. 2) that the probabilistic arguments of Rao and Tao, inspired by Alweiss, Lovett, Wu and Zhang, give this with $r(p,k)=\Theta(p\log(pk))$ (in their union-bound form, with $r(p,k)=Bp\log(pk)$); the improvement to $\Theta(p\log k)$ is the paper's.

Proof pointer

P. 2, proof of Lemma 2, with $C=4B$ for the constant $B$ of Theorem 3. Assign each element of the ground set independently to one of $2p$ classes uniformly at random. Each class is distributed as $X_\delta$ with $\delta=1/(2p)$, and Theorem 3 with $\epsilon=1/2$ applies because $r(p,k)=2Bp\log(k^2)\geq B\delta^{-1}\log(k/\epsilon)$, so each class contains a member of $\mathcal S$ with probability more than $1/2$. By linearity of expectation some partition has at least $p$ classes containing a member, and members in different classes are disjoint. The paper's point is the use of $2p$ classes and expectation in place of $p$ classes and a union bound.

Read depth

Claims checked: the definition of spread on p. 1, Lemma 2 and its proof on p. 2 were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

Theorem 3, the main technical estimate of Rao and Tao, which the paper quotes and derives in its appendix from Rao's proof.

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 step that gives Theorem 1's bound $f(n,k)\leq(Ck\log n)^n$; it does not give the $c_k^n$ bound the problem asks for.