Erdős-Rado sunflower conjecture

Statement

The sunflower conjecture is one of several variations of the conjecture of (Erdős Rado 1960, p. 86) that for each $r>2$, $f(k,r)\le C^k$ for some constant $C>0$ depending only on $r$. The conjecture remains open even for fixed low values of $r$; for example $r=3$; it is not known whether $f(k,3)\le C^k$ for some $C>0$. A 2021 paper by Alweiss, Lovett, Wu, and Zhang gives the best progress towards the conjecture, proving that $f(k,r)\le C^k$ for $C = O(r^3\log(k)\log\log(k))$. A month after the release of the first version of their paper, Rao sharpened the bound to $C=O(r\log(rk))$; the current best-known bound is $C=O(r\log k)$.