Improved bounds for sunflowers

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.