---

Statement

Setting (p. 1). A sunflower with $p$ petals is a family of $p$ sets whose pairwise intersections are all the same set, which may be empty. $\mathrm{Sun}(p,k)$ is the least natural number $s$ such that every family of at least $s$ distinct $k$-element sets contains a sunflower with $p$ petals.

Theorem 1 (p. 1, quoted). "There is a constant $C\geq 4$ such that $\mathrm{Sun}(p,k)\leq(Cp\log k)^k$ for all integers $p,k\geq 2$."

The paper places this against Rao's bound $\mathrm{Sun}(p,k)\leq(Cp\log(pk))^k$ (reproved by Tao), so the gain is the removal of the $\log p$ factor, and against the Erdős–Rado bounds $(p-1)^k<\mathrm{Sun}(p,k)\leq(p-1)^kk!+1$ (p. 1).

Proof pointer

P. 1, the paragraph after Theorem 1. With $r(p,k)=Cp\log k+\mathbb 1_{\{k=1\}}p$ the paper shows $\mathrm{Sun}(p,k)\leq r(p,k)^k$ for all $p\geq2$ and $k\geq1$ by induction on $k$. The case $k=1$ is immediate since $r(p,1)=p$. For $k\geq2$, a family that is $r(p,k)$-spread has $p$ disjoint members by Lemma 2, and these form a sunflower; otherwise some non-empty $T$ lies in more than $r(p,k)^{k-|T|}\geq r(p,k-|T|)^{k-|T|}$ members, and induction applied to those members with $T$ removed gives the sunflower.

Read depth

Claims checked: the definitions, Theorem 1 and the induction on p. 1 were read clause by clause on the page images of the print. Nothing here is independently reviewed.

Dependencies

Lemma 2, which rests on the external Theorem 3 of Rao and Tao.

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: with the problem's $n$ as the paper's $k$ and the problem's $k$ as the paper's $p$, the theorem gives $f(n,k)\leq(Ck\log n)^n$ for all $n,k\geq2$. For fixed $k$ this is $(\log n)^{n(1+o(1))}$, not the $c_k^n$ bound the problem asks for; the paper states that the conjecture remains open (p. 1).