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  "body": "\n## Statement\n\nLet $k \\ge 1$ be an integer.\nFor a [[family|0C21Q694AG]] $\\mathcal{F}$ of $k$-element sets, the *shadow* $\\partial \\mathcal{F}$ is the family of all $(k-1)$-element sets that are contained in some member of $\\mathcal{F}$.\nEvery positive integer $m$ can be written in exactly one way as\n$$m = \\binom{a_k}{k} + \\binom{a_{k-1}}{k-1} + \\cdots + \\binom{a_t}{t}$$\nwith integers $a_k \u003e a_{k-1} \u003e \\cdots \u003e a_t \\ge t \\ge 1$.\nIf $\\mathcal{F}$ is a family of $m$ sets of size $k$, then\n$$|\\partial \\mathcal{F}| \\ge \\binom{a_k}{k-1} + \\binom{a_{k-1}}{k-2} + \\cdots + \\binom{a_t}{t-1}.$$\n\n## Remarks\n\nThe bound is attained by the first $m$ sets of size $k$ of positive integers in the colexicographic order.\nIn this order, $A$ comes before $B$ when the largest element of $A \\mathbin{\\triangle} B$ lies in $B$.\nThe proof shows, with compression operators, that one may assume $\\mathcal{F}$ is an initial segment of this order, and then counts.\n\nExample: let $k = 2$ and $m = 3$, so $m = \\binom{3}{2}$ and the bound is $\\binom{3}{1} = 3$.\nA graph with $3$ edges, seen as a family of $2$-sets, covers at least $3$ vertices, and the triangle covers exactly $3$.\n\nA simpler form due to Lovász: if $|\\mathcal{F}| = \\binom{x}{k}$ for a real number $x \\ge k$, then $|\\partial \\mathcal{F}| \\ge \\binom{x}{k-1}$, where $\\binom{x}{j} = x(x-1)\\cdots(x-j+1)/j!$.\n\n## Explanations\n\n### L1 human\n\nGiven many sets of the same size, look at all the smaller sets you get by removing one element.\nThe theorem says exactly how few such smaller sets there can be.\nThe answer is always attained by packing the sets as tightly as possible, using as few elements as possible.\n\n### L2 human\n\nThe theorem characterises the possible $f$-vectors of simplicial complexes, since the faces of a complex form a family closed under taking shadows.\nIt implies the [[Erdős–Ko–Rado theorem|FJ92Q6VTJH]].\nIts numerical form, through the $k$-binomial representation of $m$, reappears in Macaulay's theorem on Hilbert functions and in the [[g-theorem|4CZMWQVC30]] for polytopes.\n",
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