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  "body": "\n## Statement\n\nLet $\\mathcal{F}$ be a finite [[family|0C21Q694AG]] of finite sets that contains at least one nonempty set.\nSuppose that $\\mathcal{F}$ is union-closed: $A \\cup B \\in \\mathcal{F}$ for all $A, B \\in \\mathcal{F}$, where $\\cup$ is the [[union|0X1GSWNBN1]].\nThen there is an element $x$ that belongs to at least $|\\mathcal{F}| / 2$ of the sets in $\\mathcal{F}$.\n\n## Remarks\n\nThe conjecture is attributed to Péter Frankl and is also called Frankl's conjecture.\nIt is open.\nIn 2022 Justin Gilmer proved that some element lies in at least $0.01 \\, |\\mathcal{F}|$ of the sets, the first bound by a positive constant fraction.\nSeveral authors then improved the constant to about $0.38$.\nThe conjecture is known when $\\mathcal{F}$ contains a set with one or two elements: such an element, or one of the two elements, works.\n\nExample: $\\mathcal{F} = \\set{\\emptyset, \\set{1}, \\set{2}, \\set{1, 2}}$ is union-closed, and the element $1$ lies in $2 = |\\mathcal{F}|/2$ of its sets, so the bound $1/2$ cannot be improved.\nThe nonempty hypothesis is needed: $\\mathcal{F} = \\set{\\emptyset}$ is union-closed, but no element lies in any of its sets.\n\n## Explanations\n\n### L1 human\n\nTake a collection of sets in which the union of any two sets of the collection is again in the collection.\nThe conjecture says that some single element appears in at least half of the sets.\nIt is very easy to state, yet it resisted all attempts for decades, and only recently was a constant fraction proved.\n",
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      "first_stated": "1979",
      "posed_by": "Péter Frankl"
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    "msc": [
      "05D05"
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    "name": "Union-closed sets conjecture",
    "same_as": {
      "wikidata": "Q2993338",
      "wikipedia": "Union-closed sets conjecture"
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