Antichain
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In the mathematical area of order theory, an antichain in a partially ordered set S is a subset A of S such that every pair of members of A is incomparable, i.e., for any x, y in A, neither x ≤ y nor y ≤ x.
Dilworth's theorem states that the non-existence of an antichain A of size n+1 in S is a necessary and sufficient condition for S to be the union of n total orders. This motivates questions about the size of maximal antichain.
Sperner's theorem says that in the power set of a finite set X of size n, ordered by inclusion, a maximal antichain has size at most <math>{n \choose \lfloor{n/2}\rfloor}<math>, in other words, the maximal antichain can be found at the "median" size subsets of X. For details see Sperner family.