Sperner's theorem is a fundamental result in extremal set theory. Let 
be an 
-element
set, and let 
be a collection of subsets of 
such that no member of 
contains another member of 
. Equivalently, 
is an antichain in the Boolean
lattice of subsets of 
ordered by inclusion. Then Sperner's
theorem states that
 |
where 
is a binomial coefficient and 
is the floor function.
The bound is sharp since the family of all 
-element subsets of 
is an antichain of this size. Equivalently, the largest antichain in the Boolean
lattice 
has size equal to the middle binomial coefficient. For even 
, the unique extremal family is the middle layer, i.e., the
rank 
layer consisting of all 
-element subsets, while for odd 
, the two middle layers have ranks 
and 
.
Sperner's theorem was proved by Sperner (1928), and should not be confused with Sperner's Lemma, a different theorem about colorings of triangulated simplices.
