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.
A family of sets in which none of the sets is a strict subset of another is called a Sperner family, or an antichain of sets, or a clutter. For example, the family of k-element subsets of an n-element set is a Sperner family. No set in this family can contain any of the others, because a containing set has to be strictly bigger than the set it contains, and in this family all sets have equal size. The value of k that makes this example have as many sets as possible is n/2 if n is even, or either of the nearest integers to n/2 if n is odd. For this choice, the number of sets in the family is .
Sperner's theorem states that these examples are the largest possible Sperner families over an n-element set. Formally, the theorem states that,
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 many proofs of Sperner's theorem, each leading to different generalizations (see Anderson (1987)).
The following proof is due to Lubell (1966). Let sk denote the number of k-sets in S. For all 0 ≤ k ≤ n,
and, thus,
Since S is an antichain, we can sum over the above inequality from k = 0 to n and then apply the LYM inequality to obtain
which means
This completes the proof of part 1.
To have equality, all the inequalities in the preceding proof must be equalities. Since
if and only if or we conclude that equality implies that S consists only of sets of sizes or For even n that concludes the proof of part 2.
For odd n there is more work to do, which we omit here because it is complicated. See Anderson (1987), pp. 3–4.
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 lengthr. An r-chain-free family (also called an r-family) is a family of subsets of E that contains no chain of length r. Erdős (1945) 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 ofE if the sets form a partition of E. Meshalkin (1963) 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.
Beck & Zaslavsky (2002) 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. Rota & Harper (1971) 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.
Beck & Zaslavsky (2003) obtained a Meshalkin-like generalization of the Rota–Harper theorem. In PG(d, Fq), a Meshalkin sequence of length p is a sequence of projective subspaces such that no proper subspace of PG(d, Fq) contains them all and their dimensions sum to . The theorem is that a family of Meshalkin sequences of length p in PG(d, Fq), such that the subspaces appearing in position i of the sequences contain no chain of length r for each is not more than the sum of the largest of the quantities
where (in which we assume that ) denotes the p-Gaussian coefficient
and
the second elementary symmetric function of the numbers
In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".
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In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety.
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
In combinatorial mathematics, the Lubell–Yamamoto–Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by Bollobás (1965), Lubell (1966), Meshalkin (1963), and Yamamoto (1954). It is named for the initials of three of its discoverers. To include the initials of all four discoverers, it is sometimes referred to as the YBLM inequality.
In combinatorics, a Sperner family, or clutter, is a family F of subsets of a finite set E in which none of the sets contains another. Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of E. A Sperner family is also sometimes called an independent system or irredundant set.
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