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  "body": "\n## Statement\n\nA *sunflower* with $p$ petals is a [[family|0C21Q694AG]] of $p$ distinct sets $S_1, \\ldots, S_p$ whose pairwise [[intersections|0HBRW4E6G1]] are all equal.\nThat is, there is a set $Y$, the core, with $S_i \\cap S_j = Y$ for all $i \\ne j$.\nLet $k \\ge 1$ and $p \\ge 1$ be integers.\nLet $\\mathcal{F}$ be a family of sets, each with exactly $k$ elements, such that $|\\mathcal{F}| \u003e k! \\, (p - 1)^k$.\nThen $\\mathcal{F}$ contains a sunflower with $p$ petals.\n\n## Remarks\n\nThe proof is by induction on $k$.\nTake a maximal subfamily of pairwise disjoint sets.\nIf it has at least $p$ members, they form a sunflower with empty core.\nOtherwise their union $U$ has at most $(p - 1) k$ elements and meets every set of $\\mathcal{F}$, so some $x \\in U$ lies in more than $(k-1)! \\, (p-1)^{k-1}$ sets.\nRemove $x$ from these sets and apply the induction hypothesis to get a sunflower, then add $x$ back to its core.\n\nExample: for $k = 1$ and $p = 3$, a family of $3$ distinct one-element sets is a sunflower with empty core, and $3 \u003e 1! \\cdot 2^1$.\nThe result is due to Erdős and Rado, who conjectured that $k! \\, (p-1)^k$ can be replaced by $C_p^k$ for a constant $C_p$ that depends only on $p$; this is open.\nIn 2019 Alweiss, Lovett, Wu and Zhang improved the bound to $(C p \\log k \\log \\log k)^k$ for an absolute constant $C$, and later work lowered it to $(C p \\log k)^k$.\n\n## Explanations\n\n### L1 human\n\nA sunflower is a group of sets that all overlap in exactly the same core, like petals around the centre of a flower.\nThe lemma says that any large enough collection of sets of the same size must contain a sunflower with as many petals as you like.\nSo size alone forces this very regular pattern.\n\n### L2 human\n\nThe sunflower lemma is a basic tool in extremal set theory and in theoretical computer science, for example in lower bounds for monotone circuits.\nIts proof is a typical Ramsey-type argument: either many sets are disjoint, or many share an element, and then one recurses.\nIt belongs to the circle of results of [[Ramsey theory|G5TWXV565S]], where a large structure always contains an orderly part.\n",
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