WOW! The new paper https://arxiv.org/abs/1908.08483 improved bounds for the sunflower lemma gives the most dramatic progress on the sunflower conjecture since it was asked. Congratulations to Ryan Alweiss, Shachar Lovett, Kewen Wu, Jiapeng Zhang.
(Written on my smartphone will expand it when reconnected to my laptop.) (Reconnected) Rather, I will write about it again in a few weeks. Let me mention now that while the old difficult and ingenious improvements stayed in the neighborhood of Erdos and Rado initial upper bound the new result is in the neighborhood of the conjecture! (And is tight for a certain robust version of the problem!)
Update: Let me mention an important progress on the sunflower conjecture from 2018 by Junichiro Fukuyama in his paper Improved Bound on Sets Including No Sunflower with Three Petals. I missed Fukuyama’s paper at the time and I thank Sasha Kostochka and Andrew Thomason for telling me about it now. Further update: It seems that Junichiro’s argument (while insightful) has a gap.
Updates: (from comments below) Anup Rao presents a simplification of the original Alweiss, Lovett, Wu, and Zhang argument. https://arxiv.org/abs/1909.04774; An excellent Quanta Magazine article by Kevin Hartnett https://www.quantamagazine.org/mathematicians-begin-to-tame-wild-sunflower-problem-20191021/ . Terry Tao posted a version of Rao’s proof