Improved bounds for the sunflower lemma

A sunflower with $r$ petals is a collection of sets so that the intersection each pair equal to all them. Erd?s and Rado proved lemma: for any fixed $r$, family size $w$, at least about $w^w$ sets, must contain petals. The famous conjecture states bound on number can be improved $c^w$ some constant $c$. In this paper, we improve $\mathrm{log}\ w)^w$. fact, prove result robust notion sunflowers, which obtain sharp up lower order terms.