Sperner's theorem

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, and 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.

Statement

A family of sets that does not include two sets X and Y for which X ⊂ Y is called a Sperner family. 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 the nearest integer to n/2 if n is odd. For this choice, the number of sets in the family is ${\tbinom {n}{\lfloor n/2\rfloor }}$ .

Sperner's theorem states that these examples are the largest possible Sperner families over an n-element set. Formally, the theorem states that, for every Sperner family S whose union has a total of n elements,

|S|\leq {\binom {n}{\lfloor n/2\rfloor }}.

Partial orders

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 ${\binom {n}{\lfloor n/2\rfloor }}$ .

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.

Proof

The following proof is due to Lubell (1966). Let s_k denote the number of k-sets in S. For all 0 ≤ k ≤ n,

{n \choose \lfloor {n/2}\rfloor }\geq {n \choose k}

and, thus,

{s_{k} \over {n \choose \lfloor {n/2}\rfloor }}\leq {s_{k} \over {n \choose k}}.

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

\sum _{{k=0}}^{n}{s_{k} \over {n \choose \lfloor {n/2}\rfloor }}\leq \sum _{{k=0}}^{n}{s_{k} \over {n \choose k}}\leq 1,

which means

|S|=\sum _{{k=0}}^{n}s_{k}\leq {n \choose \lfloor {n/2}\rfloor }.

This completes the proof.

Generalizations

There are several generalizations of Sperner's theorem for subsets of ${\mathcal P}(E),$ the poset of all subsets of E.

No long chains

A chain is a subfamily $\{S_{0},S_{1},\dots ,S_{r}\}\subseteq {\mathcal P}(E)$ that is totally ordered, i.e., $S_{0}\subset S_{1}\subset \dots \subset S_{r}$ (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. Erdős (1945) proved that the largest size of an r-chain-free family is the sum of the r largest binomial coefficients ${\binom {n}{i}}$ . The case r = 1 is Sperner's theorem.

p-compositions of a set

In the set ${\mathcal P}(E)^{p}$ of p-tuples of subsets of E, we say a p-tuple $(S_{1},\dots ,S_{p})$ is ≤ another one, $(T_{1},\dots ,T_{p}),$ if $S_{i}\subseteq T_{i}$ for each i = 1,2,...,p. We call $(S_{1},\dots ,S_{p})$ a p-composition of E if the sets $S_{1},\dots ,S_{p}$ form a partition of E. Meshalkin (1963) proved that the maximum size of an antichain of p-compositions is the largest p-multinomial coefficient ${\binom {n}{n_{1}\ n_{2}\ \dots \ n_{p}}},$ that is, the coefficient in which all n_i 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 $S_{2}=E\setminus S_{1}$ and the assumptions reduce to the sets $S_{1}$ being a Sperner family.

No long chains in p-compositions of a set

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 $i=1,2,\dots ,p-1$ (but not necessarily for i = p), is not greater than the sum of the $r^{{p-1}}$ largest p-multinomial coefficients.

Projective geometry analog

In the finite projective geometry PG(d, F_q) of dimension d over a finite field of order q, let ${\mathcal L}(p,F_{q})$ 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 ${\mathcal L}(p,F_{q})$ is the largest Gaussian coefficient ${\begin{bmatrix}d+1\\k\end{bmatrix}};$ 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 ${\mathcal L}(p,F_{q})$ is the sum of the r largest Gaussian coefficients. Their proof is by a projective analog of the LYM inequality.

No long chains in p-compositions of a projective space

Beck & Zaslavsky (2003) obtained a Meshalkin-like generalization of the Rota–Harper theorem. In PG(d, F_q), a Meshalkin sequence of length p is a sequence $(A_{1},\ldots ,A_{p})$ of projective subspaces such that no proper subspace of PG(d, F_q) contains them all and their dimensions sum to $d-p+1$ . The theorem is that a family of Meshalkin sequences of length p in PG(d, F_q), such that the subspaces appearing in position i of the sequences contain no chain of length r for each $i=1,2,\dots ,p-1,$ is not more than the sum of the largest $r^{{p-1}}$ of the quantities

{\begin{bmatrix}d+1\\n_{1}\ n_{2}\ \dots \ n_{p}\end{bmatrix}}q^{{s_{2}(n_{1},\ldots ,n_{p})}},

where ${\begin{bmatrix}d+1\\n_{1}\ n_{2}\ \dots \ n_{p}\end{bmatrix}}$ (in which we assume that $d+1=n_{1}+\cdots +n_{p}$ ) denotes the p-Gaussian coefficient

{\begin{bmatrix}d+1\\n_{1}\end{bmatrix}}{\begin{bmatrix}d+1-n_{1}\\n_{2}\end{bmatrix}}\cdots {\begin{bmatrix}d+1-(n_{1}+\cdots +n_{{p-1}})\\n_{p}\end{bmatrix}}

and

s_{2}(n_{1},\ldots ,n_{p}):=n_{1}n_{2}+n_{1}n_{3}+n_{2}n_{3}+n_{1}n_{4}+\cdots +n_{{p-1}}n_{p},

the second elementary symmetric function of the numbers $n_{1},n_{2},\dots ,n_{p}.$

References

Anderson, Ian (1987), "Sperner's theorem", Combinatorics of Finite Sets, Oxford University Press, pp. 2–4 .
Beck, Matthias; Zaslavsky, Thomas (2002), "A shorter, simpler, stronger proof of the Meshalkin-Hochberg-Hirsch bounds on componentwise antichains", Journal of Combinatorial Theory, Series A, 100 (1): 196–199, doi:10.1006/jcta.2002.3295, MR 1932078 .
Beck, Matthias; Zaslavsky, Thomas (2003), "A Meshalkin theorem for projective geometries", Journal of Combinatorial Theory, Series A, 102 (2): 433–441, doi:10.1016/S0097-3165(03)00049-9, MR 1979545 .
Engel, Konrad (1997), Sperner theory, Encyclopedia of Mathematics and its Applications, 65, Cambridge: Cambridge University Press, p. x+417, doi:10.1017/CBO9780511574719, ISBN 0-521-45206-6, MR 1429390 .
Engel, K. (2001), "Sperner theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Erdős, P. (1945), "On a lemma of Littlewood and Offord" (PDF), Bulletin of the American Mathematical Society, 51: 898–902, doi:10.1090/S0002-9904-1945-08454-7, MR 0014608
Lubell, D. (1966), "A short proof of Sperner's lemma", Journal of Combinatorial Theory, 1 (2): 299, doi:10.1016/S0021-9800(66)80035-2, MR 0194348 .
Meshalkin, L.D. (1963), "Generalization of Sperner's theorem on the number of subsets of a finite set. (In Russian)", Theory of Probability and its Applications, 8 (2): 203–204, doi:10.1137/1108023 .
Rota, Gian-Carlo; Harper, L. H. (1971), "Matching theory, an introduction", Advances in Probability and Related Topics, Vol. 1, New York: Dekker, pp. 169–215, MR 0282855 .
Sperner, Emanuel (1928), "Ein Satz über Untermengen einer endlichen Menge", Mathematische Zeitschrift (in German), 27 (1): 544–548, doi:10.1007/BF01171114, JFM 54.0090.06 .

External links

Sperner's Theorem at cut-the-knot
Sperner's theorem on the polymath1 wiki

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