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  "body": "\n## Statement\n\nLet $\\mathcal{F} = (S_1, \\dots, S_n)$ be a finite family of finite sets.\nA transversal of $\\mathcal{F}$ is an injective function $f : \\{1, \\dots, n\\} \\to \\bigcup_i S_i$ with $f(i) \\in S_i$ for every $i$.\nThen $\\mathcal{F}$ has a transversal if and only if $\\left| \\bigcup_{i \\in I} S_i \\right| \\ge |I|$ for every $I \\subseteq \\{1, \\dots, n\\}$.\n\n## Remarks\n\nThe condition on all subfamilies is the marriage condition. It is necessary, because a transversal $f$ maps $I$ injectively into $\\bigcup_{i \\in I} S_i$.\nA transversal is also called a system of distinct representatives. The sets $S_i$ may repeat in the family.\nThe theorem is equivalent to [[Hall's theorem for bipartite graphs|0X2GJPNPFN]].\n\n## History\n\nPhilip Hall proved the theorem in 1935.\nThe name marriage theorem comes from a 1950 paper by Halmos and Vaughan.\n\n## Explanations\n\n### L1 human\n\nEach of $n$ people has a list of acceptable partners, and we want to give each person a different partner from the list.\nThis is possible exactly when every group of $k$ people has at least $k$ names on their lists together.\n",
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