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  "body": "\n## Statement\n\nThe **sunflower conjecture** is one of several variations of the conjecture of (Erdős Rado 1960, p. 86) that for each $r\u003e2$, $f(k,r)\\le C^k$ for some constant $C\u003e0$ 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 $C\u003e0$. 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 $C=O(r\\log k)$.\n",
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