Note on Sunflowers

Résumé : A sunflower with p petals consists of p sets whose pairwise intersections are identical. Building upon a breakthrough of Alweiss, Lovett, Wu, and Zhang from 2019, Rao proved that any family of (Cp\log(pk))^k distinct k-element sets contains a sunflower with p petals, where C>0 is a constant; this bound was reproved by Tao. In this note we record that, by a minor variant of their probabilistic arguments, any family of (Cp\log k)^k distinct k-element sets contain a sunflower with p petals, where C>0 is a constant.