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  "body": "\n## Statement\n\nLet $E$ be a set with $n$ elements.\nLet $\\mathcal{S}$ be a family of subsets of $E$ in which no set is a proper subset of another.\nThen $|\\mathcal{S}| \\le \\binom{n}{\\lfloor n/2 \\rfloor}$.\nEquality holds if and only if $\\mathcal{S}$ is the family of all subsets of size $\\lfloor n/2 \\rfloor$, or of all subsets of size $\\lceil n/2 \\rceil$.\n\n## Remarks\n\nA family with this property is a Sperner family, or an antichain of sets.\nFor each $k$, the subsets of size $k$ form an antichain. The [[binomial coefficient|092GEF38YD]] $\\binom{n}{k}$ is largest for $k = \\lfloor n/2 \\rfloor$.\nLubell gave a short proof in 1966, with the Lubell–Yamamoto–Meshalkin inequality.\n\n## History\n\nEmanuel Sperner published the theorem in 1928.\n\n## Explanations\n\n### L1 human\n\nYou want many subsets of a set, but no one of them may contain another.\nThe best you can do is to take all subsets of half the size.\n",
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