Спернер теоремасы және жиын теориясының экстремалды мәселелері

the theorem about simplexes
Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set theory. It is named after Emanuel Sperner, who published it in 1928. This result is sometimes called Sperner's lemma, but the name "Sperner's lemma" also refers to an unrelated result on coloring triangulations. To differentiate the two results, the result on the size of a Sperner family is now more commonly known as Sperner's theorem.

Sperner's theorem can also be stated in terms of partial order width. The family of all subsets of an n element set (its power set) can be partially ordered by set inclusion; in this partial order, two distinct elements are said to be incomparable when neither of them contains the other. The width of a partial order is the largest number of elements in an antichain, a set of pairwise incomparable elements. Translating this terminology into the language of sets, an antichain is just a Sperner family, and the width of the partial order is the maximum number of sets in a Sperner family. Thus, another way of stating Sperner's theorem is that the width of the inclusion order on a power set is
A graded partially ordered 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.

There are several generalizations of Sperner's theorem for subsets of the poset of all subsets of E.

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.

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.

combined the Erdös and Meshalkin theorems by adapting Meshalkin's proof of his generalized LYM inequality. They showed that the largest size of a family of p compositions such that the sets in the i th position of the p tuples, ignoring duplications, are r chain free, for every (but not necessarily for i = p), is not greater than the sum of the largest p multinomial coefficients.

In the finite projective geometry PG(d, Fq) of dimension d over a finite field of order q, let be the family of all subspaces. When partially ordered by set inclusion, this family is a lattice. proved that the largest size of an antichain in is the largest Gaussian coefficient this is the projective geometry analog, or q analog, of Sperner's theorem. They further proved that the largest size of an r chain free family in is the sum of the r largest Gaussian coefficients. Their proof is by a projective analog of the LYM inequality.