Sperner's theorem

In an n-element set, any family of subsets in which no member contains another has size at most the central binomial coefficient, \(\binom{n}{\lfloor n/2\rfloor}\). Equality is attained by all subsets of size \(\lfloor n/2\rfloor\) or \(\lceil n/2\rceil\).

Antichain: A family of elements in a partially ordered set in which no two distinct elements are comparable. Sperner's theorem bounds antichains in the subset order.
Partially ordered set: A set equipped with a relation that is reflexive, antisymmetric, and transitive. Sperner's theorem is an extremal result about a particular partially ordered set.
Emanuel Sperner: A German mathematician who proved the subset-antichain theorem published in 1928. His name is attached to the theorem and its original proof.
Sperner family: A family of subsets in which no distinct member contains another. Such families are precisely the objects whose maximum size the theorem determines.
Erdős–Ko–Rado theorem: A theorem bounding intersecting families of fixed-size subsets when the ground set is sufficiently large. It adds an intersection constraint, unlike Sperner's incomparability condition.
Boolean lattice: The partially ordered set of all subsets of a finite set, ordered by inclusion. The theorem concerns antichains in this lattice.
Subset: A set whose every element belongs to another specified set. The theorem's objects are subsets ordered by containment.
Extremal set theory: The study of largest or smallest families of sets satisfying specified constraints. Sperner's theorem is a foundational maximum-size result in this field.
Monotone Boolean function: A Boolean function whose value cannot decrease when any input coordinate changes from zero to one. Its minimal true inputs form an antichain bounded by Sperner's theorem.
Dilworth's theorem: A theorem equating the largest antichain size in a finite poset with its minimum chain-cover size. It characterizes antichain width through chain partitions rather than giving a Boolean-lattice rank bound.

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