Let . A family of subsets of that are pairwise unrelated with respect to inclusion is called a Sperner family (or Sperner system) on . Examples are the families
Since the binomial coefficients satisfy the inequalities
in these examples , if is even, and as well as , if is odd, have maximum size. Sperner's theorem from 1928 states that these best examples have even maximum size among all Sperner families on and that they are the only optimal families.
Given a Sperner family , let and . In his original proof, E. Sperner used a shifting technique: Consider the smallest with and replace by its upper shadow
Double-counting easily yields and, equivalently,
Thus, each Sperner family can be shifted from below to the "middle" and, analogously, from above to the "middle" and thereby increasing its size.
The inequality (a1) holds for all and all , and this property is called the normalized matching property of the lattice of subsets of . If and are fixed, the best possible estimate of the upper shadow, and, dually, of the lower shadow (replace by and superset by subset), is given by the Kruskal–Katona theorem.
Sperner's theorem follows also easily from the inequality
which can be obtained by counting in two different ways the number of pairs where , is a permutation of and . This inequality was proved independently by D. Lubell, S. Yamamoto and L. Meshalkin, and is hence called the LYM inequality; a more general form of it was given by B. Bollobás.
An essential part of Sperner theory consists of the study of other partially ordered sets having analogous properties, e.g. LYM posets and Peck posets (cf. Sperner property).
Details can be found in [a1].
|[a1]||K. Engel, "Sperner theory" , Cambridge Univ. Press (1997)|
|[a2]||E. Sperner, "Ein Satz über Untermengen einer endlichen Menge" Math. Z. , 27 (1928) pp. 544–548|
Sperner theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sperner_theorem&oldid=12799