Note on Sunflowers

Abstract: 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.