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| '''Sperner's theorem''', in [[discrete mathematics]], describes the largest possible [[family of sets|families]] of [[finite set]]s none of which contain any other sets in the family. It is one of the central results in [[Extremal combinatorics|extremal set theory]], and is named after [[Emanuel Sperner]], who published it in 1928.
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| 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.
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| ==Statement==
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| 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 <math>\tbinom{n}{\lfloor n/2\rfloor}</math>.
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| Sperner's theorem states that these examples are the largest possible Sperner families over an ''n''-element set.
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| Formally, the theorem states that, for every Sperner family ''S'' whose union has a total of ''n'' elements,
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| :<math>|S| \le \binom{n}{\lfloor n/2\rfloor}.</math>
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| ==Partial orders==
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| 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.
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| Thus, another way of stating Sperner's theorem is that the width of the inclusion order on a power set is <math>\binom{n}{\lfloor n/2\rfloor}</math>.
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| A [[Graded poset|graded]] [[partially ordered set]] is said to have the [[Sperner property of a partially ordered set|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. | |
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| == Proof ==
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| The following proof is due to {{harvtxt|Lubell|1966}}. Let ''s<sub>k</sub>'' denote the number of ''k''-sets in ''S''. For all 0 ≤ ''k'' ≤ ''n'',
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| :<math>{n \choose \lfloor{n/2}\rfloor} \ge {n \choose k}</math>
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| and, thus,
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| :<math>{s_k \over {n \choose \lfloor{n/2}\rfloor}} \le {s_k \over {n \choose k}}.</math>
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| Since ''S'' is an antichain, we can sum over the above inequality from ''k'' = 0 to ''n'' and then apply the [[Lubell–Yamamoto–Meshalkin inequality|LYM inequality]] to obtain
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| :<math>\sum_{k=0}^n{s_k \over {n \choose \lfloor{n/2}\rfloor}} \le \sum_{k=0}^n{s_k \over {n \choose k}} \le 1,</math>
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| which means
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| :<math> |S| = \sum_{k=0}^n s_k \le {n \choose \lfloor{n/2}\rfloor}.</math>
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| This completes the proof.
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| ==Generalizations==
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| There are several generalizations of Sperner's theorem for subsets of <math>\mathcal P(E),</math> the poset of all subsets of ''E''.
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| ===No long chains===
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| A '''chain''' is a subfamily <math>\{S_0,S_1,\dots,S_r\} \subseteq \mathcal P(E)</math> that is totally ordered, i.e., <math>S_0 \subset S_1\subset \dots\subset S_r</math> (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''. {{harvtxt|Erdős|1945}} proved that the largest size of an ''r''-chain-free family is the sum of the ''r'' largest binomial coefficients <math>\binom{n}{i}</math>. The case ''r'' = 1 is Sperner's theorem.
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| ===''p''-compositions of a set===
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| In the set <math>\mathcal P(E)^p</math> of ''p''-tuples of subsets of ''E'', we say a ''p''-tuple <math>(S_1,\dots,S_p)</math> is ≤ another one, <math>(T_1,\dots,T_p),</math> if <math>S_i \subseteq T_i</math> for each ''i'' = 1,2,...,''p''. We call <math>(S_1,\dots,S_p)</math> a ''p'''''-composition of''' ''E'' if the sets <math>S_1,\dots,S_p</math> form a partition of ''E''. {{harvtxt|Meshalkin|1963}} proved that the maximum size of an antichain of ''p''-compositions is the largest ''p''-multinomial coefficient <math>\binom{n}{n_1\ n_2\ \dots\ n_p},</math> that is, the coefficient in which all ''n''<sub>''i''</sub> are as nearly equal as possible (i.e., they differ by at most 1). Meshalkin proved this by proving a generalized LYM inequality.
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| The case ''p'' = 2 is Sperner's theorem, because then <math>S_2 = E \setminus S_1</math> and the assumptions reduce to the sets <math>S_1</math> being a Sperner family.
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| ===No long chains in ''p''-compositions of a set===
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| {{harvtxt|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 <math>i = 1,2,\dots,p-1</math> (but not necessarily for ''i'' = ''p''), is not greater than the sum of the <math>r^{p-1}</math> largest ''p''-multinomial coefficients.
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| ===Projective geometry analog===
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| In the finite projective geometry PG(''d'', ''F''<sub>''q''</sub>) of dimension ''d'' over a finite field of order ''q'', let <math>\mathcal L(p,F_q)</math> be the family of all subspaces. When partially ordered by set inclusion, this family is a lattice. {{harvtxt|Rota|Harper|1971}} proved that the largest size of an antichain in <math>\mathcal L(p,F_q)</math> is the largest [[Gaussian coefficient]] <math>\begin{bmatrix} d+1 \\ k\end{bmatrix};</math> this is the projective-geometry analog, or ''q'''''-analog''', of Sperner's theorem. | |
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| They further proved that the largest size of an ''r''-chain-free family in <math>\mathcal L(p,F_q)</math> is the sum of the ''r'' largest Gaussian coefficients. Their proof is by a projective analog of the LYM inequality.
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| ===No long chains in ''p''-compositions of a projective space===
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| {{harvtxt|Beck|Zaslavsky|2003}} obtained a Meshalkin-like generalization of the Rota–Harper theorem. In PG(''d'', ''F''<sub>''q''</sub>), a '''Meshalkin sequence''' of length ''p'' is a sequence <math>(A_1,\ldots,A_p)</math> of projective subspaces such that no proper subspace of PG(''d'', ''F''<sub>''q''</sub>) contains them all and their dimensions sum to <math>d-p+1</math>. The theorem is that a family of Meshalkin sequences of length ''p'' in PG(''d'', ''F''<sub>''q''</sub>), such that the subspaces appearing in position ''i'' of the sequences contain no chain of length ''r'' for each <math>i = 1,2,\dots,p-1,</math> is not more than the sum of the largest <math>r^{p-1}</math> of the quantities
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| :<math>\begin{bmatrix} d+1 \\ n_1\ n_2\ \dots\ n_p \end{bmatrix} q^{s_2(n_1,\ldots,n_p)},</math>
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| where <math>\begin{bmatrix} d+1 \\ n_1\ n_2\ \dots\ n_p \end{bmatrix}</math> (in which we assume that <math>d+1 = n_1+\cdots+n_p</math>) denotes the ''p''-Gaussian coefficient
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| :<math>\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}</math>
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| and
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| :<math>s_2(n_1,\ldots,n_p) := n_1n_2 + n_1n_3 + n_2n_3 + n_1n_4 + \cdots + n_{p-1}n_p,</math>
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| the second [[elementary symmetric function]] of the numbers <math>n_1, n_2, \dots, n_p.</math>
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| ==References==
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| *{{citation
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| | last = Anderson | first = Ian
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| | title = Combinatorics of Finite Sets
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| | publisher = Oxford University Press
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| | year = 1987
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| | contribution = Sperner's theorem
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| | pages = 2–4}}.
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| *{{citation
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| | last1 = Beck | first1 = Matthias
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| | last2 = Zaslavsky | first2 = Thomas | author2-link = Thomas Zaslavsky
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| | doi = 10.1006/jcta.2002.3295
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| | issue = 1
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| | journal = [[Journal of Combinatorial Theory]]
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| | mr = 1932078
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| | pages = 196–199
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| | series = Series A
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| | title = A shorter, simpler, stronger proof of the Meshalkin-Hochberg-Hirsch bounds on componentwise antichains
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| | volume = 100
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| | year = 2002}}.
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| *{{citation
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| | last1 = Beck | first1 = Matthias
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| | last2 = Zaslavsky | first2 = Thomas | author2-link = Thomas Zaslavsky
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| | doi = 10.1016/S0097-3165(03)00049-9
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| | issue = 2
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| | journal = [[Journal of Combinatorial Theory]]
| |
| | mr = 1979545
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| | pages = 433–441
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| | series = Series A
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| | title = A Meshalkin theorem for projective geometries
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| | volume = 102
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| | year = 2003}}.
| |
| *{{citation
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| | last = Engel | first = Konrad
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| | doi = 10.1017/CBO9780511574719
| |
| | isbn = 0-521-45206-6
| |
| | location = Cambridge
| |
| | mr = 1429390
| |
| | page = x+417
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| | publisher = Cambridge University Press
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| | series = Encyclopedia of Mathematics and its Applications
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| | title = Sperner theory
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| | volume = 65
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| | year = 1997}}.
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| *{{springer |first=K. |last=Engel |title=Sperner theorem |id=S/s130500}}
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| *{{citation
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| | last = Erdős | first = P. | authorlink = Paul Erdős
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| | doi = 10.1090/S0002-9904-1945-08454-7
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| | journal = [[Bulletin of the American Mathematical Society]]
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| | mr = 0014608
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| | pages = 898–902
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| | title = On a lemma of Littlewood and Offord
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| | url = http://www.renyi.hu/~p_erdos/1945-04.pdf
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| | volume = 51
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| | year = 1945}}
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| *{{citation
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| | last = Lubell | first = D.
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| | year = 1966
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| | title = A short proof of Sperner's lemma
| |
| | journal = [[Journal of Combinatorial Theory]]
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| | volume = 1 | issue = 2 | page = 299
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| | mr = 0194348
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| | doi = 10.1016/S0021-9800(66)80035-2}}.
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| *{{citation
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| | title = Generalization of Sperner's theorem on the number of subsets of a finite set. (In Russian)
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| | last = Meshalkin | first = L.D.
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| | journal = Theory of Probability and its Applications
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| | volume = 8
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| | issue = 2
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| | pages = 203–204
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| | year = 1963}}.
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| *{{citation
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| | last1 = Rota | first1 = Gian-Carlo | author1-link = Gian-Carlo Rota
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| | last2 = Harper | first2 = L. H.
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| | contribution = Matching theory, an introduction
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| | location = New York
| |
| | mr = 0282855
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| | pages = 169–215
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| | publisher = Dekker
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| | title = Advances in Probability and Related Topics, Vol. 1
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| | year = 1971}}.
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| *{{citation
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| | last = Sperner | first = Emanuel | authorlink = Emanuel Sperner
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| | title = Ein Satz über Untermengen einer endlichen Menge
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| | journal = [[Mathematische Zeitschrift]]
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| | volume = 27 | issue = 1 | year = 1928
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| | doi = 10.1007/BF01171114
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| | language = German
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| | pages = 544–548 | jfm = 54.0090.06 }}.
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| ==External links==
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| * [http://www.cut-the-knot.org/pigeonhole/sperner.shtml Sperner's Theorem] at [[cut-the-knot]]
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| * [http://michaelnielsen.org/polymath1/index.php?title=Sperner%27s_theorem Sperner's theorem] on the polymath1 wiki
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| | |
| [[Category:Set families]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Articles containing proofs]]
| |
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This article focuses on the genetics, and the different types of androgen insensitivity syndrome. Other advantages of the Size - Genetics Extender is that it promotes overall male sexual health. However, when I talk with my girlfriends about penis pills, they all express great concern about the health hazards and they all say they would much rather their guys didn't take these pills - regardless of any supposed or perceived benefits. Imagine, waking up tomorrow morning with your penis magically 2 inches longer and a half-inch wider. In addition, men experience a better erection and an increase in sexual stamina.
It relies on the principle that as cells within an area exposed to frequent motion, they tend to divide and multiply. Even if these methods worked, why would you want a 10 inch penis that will not get erect. Spiritualists believe that people always seemed to be "living in" the past or in the future but don't know how to live 100 percent in the present, or what we call 'the now. Second, the thickness of the penis will also be improved, which is a real plus especially for men who want to provide mutual pleasure to both themselves and their partners. Sure, you could cut, dye, or style hair, or you can work out to get a better physique - but when it comes to things like penis size, at one point, there wasn't much that men who wished for a bigger penis could do.
However, it is still not reported if the pleasure from orgasm is less or same. Your sex life can become the stuff of legends, but only by using natural male enhancements. Some use these products just to test the claims of marketers. The herbal product is advisable as they are free of side effects and can be used by anyone. This process allows the penis chambers to hold more blood than it could before.
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