Note on sunflowers

A sunflower with p petals consists of p sets whose pairwise intersections are identical. The goal of the sunflower problem is to find the smallest r = r(p, k) such that any family of r(k) distinct k-element sets contains a sunflower with p petals. Building upon a breakthrough of Alweiss, Lovett, Wu and Zhang from 2019, Rao proved that r = O(p log(pk)) suffices; this bound was reproved by Tao in 2020. In this short note we record that r = O(p log k) suffices, by using a minor variant of the probabilistic part of these recent proofs. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

The sunflower problem aims to find the smallest r such that any family of r distinct k-element sets contains a sunflower with p petals. It has been shown that r = O(p log k) suffices by using a minor variant of recent proofs.