An r-sunflower is a collection of r sets so that the intersection of any two are the same. Given a fixed constant r, how many sets of size w can we have so that no r of them form an r-sunflower? Erdos and Rado introduced this problem in 1960 and proved a bound of w^(w(1+o(1)), and until recently the best known bound was still of this form. Furthermore, Erdos offered $1000 for a proof of a bound of c^w, where c depends on r. We prove a bound of (log w)^(w(1+o(1)).

Joint work with Shachar Lovett, Kewen Wu, and Jiapeng Zhang.