set is said to have the Sperner property when one of its largest antichains is formed by a set of elements that all have the same rank. In this terminology, Sperner's theorem states that the partially ordered set of all subsets of a finite set, partially ordered by set inclusion, has the Sperner property. Proof There are many proofs of Sperner's theorem, each leading to different generalizations (see ). The following proof is due to . Let sk denote the number of k-sets in S. For all 0 ≤ k ≤ n, and, thus, Since S is an antichain, we can sum over the above inequality from k = 0 to n and then apply the LYM inequality to obtain which means This completes the proof of part 1. To have equality, all the inequalities in the preceding proof must be equalities. Since if and only if or we conclude that equality implies that S consists only of sets of sizes or For even n that concludes the proof of part 2. For odd n there is more work to do, which we omit here because it is complicated. See , pp. 3–4. Generalizations There are several generalizations of Sperner's theorem for subsets of the poset of all subsets of E. No long chains A chain is a subfamily that is totally ordered, i.e., (possibly after renumbering). The chain has r + 1 members and length r. An r-chain-free family (also called an r-family) is a family of subsets of E that contains no chain of length r. proved that the largest size of an r-chain-free family is the sum of the r largest binomial coefficients . The case r = 1 is Sperner's theorem. p-compositions of a set In the set of p-tuples of subsets of E, we say a p-tuple is ≤ another one, if for each i = 1,2,...,p. We call a p-composition of E if the sets form a partition of E. proved that the maximum size of an antichain of p-compositions is the largest p-multinomial coefficient that is, the coefficient in which all ni are as nearly equal as possible (i.e., they differ by at most 1). Meshalkin proved this by proving a generalized LYM inequality. The case p = 2 is Sperner's theorem, because then and the assumptions reduce to the sets being a Sperner family. No long chains in p-compositions of a set