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  "body": "\n## Definition\n\nLet $P$ be a finite [[graded poset|1C7TKGTB2F]] with rank function $\\rho$, and for each integer $i \\ge 0$ let $P_i = \\set{x \\in P : \\rho(x) = i}$ be the $i$-th rank level.\nThe poset $P$ has the *Sperner property* when every [[antichain|03HF498P1J]] $A \\subseteq P$ satisfies $\\abs{A} \\le \\max_i \\abs{P_i}$.\n\n## Remarks\n\nSince each $P_i$ is an antichain, the condition says that the [[width|RA5SQ908G1]] of $P$ equals $\\max_i \\abs{P_i}$.\n\nExample: the lattice of subsets of $\\set{1, \\ldots, n}$, ranked by size, has the Sperner property.\nCheck: by Sperner's theorem, an antichain has at most $\\binom{n}{\\floor{n/2}}$ members, which is the size of the middle level.\n\nNon-example: let $P$ have elements $a_1, a_2, b$ of rank $0$ and $c, d_1, d_2$ of rank $1$, with the relations $a_1 \u003c c$, $a_2 \u003c c$, $b \u003c d_1$, $b \u003c d_2$ and no others.\nEach level has $3$ elements, but $\\set{a_1, a_2, d_1, d_2}$ is an antichain with $4$ elements.\n\nA poset is strongly Sperner when, for every $k$, the largest union of $k$ antichains is no larger than the largest union of $k$ rank levels.\n",
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