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  "body": "\n## Summary\n\nA finite [[graded poset|1C7TKGTB2F]] has the Sperner property when no [[antichain|03HF498P1J]] is larger than its largest rank level.\nEach rank level is an antichain, so the property says that the [[width|RA5SQ908G1]] of the poset equals the size of a largest level.\n[[Sperner's theorem|0J87V9K1GE]] is the statement that the lattice of subsets of a finite set has this property.\n\n## Explanations\n\n### L1 human\n\nSort the elements of a graded poset into levels by rank.\nElements on one level are never comparable, so each level is an antichain.\nThe poset is Sperner when you cannot beat the biggest level by picking incomparable elements from several levels at once.\nFor the subsets of $\\set{1, 2, 3, 4}$, the biggest level has the $6$ two-element subsets, and no family of $7$ subsets avoids all inclusions.\n",
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