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| :''You may be looking for [[Sperner's theorem]] on set families''
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| In [[mathematics]], '''Sperner's lemma''' is a [[combinatorics|combinatorial]] [[analogy|analog]] of the [[Brouwer fixed point theorem]], which follows from it. Sperner's lemma states that every '''Sperner coloring''' (described below) of a [[triangulation (geometry)|triangulation]] of an ''n''-dimensional [[simplex]] contains a cell colored with a complete set of colors. The initial result of this kind was proved by [[Emanuel Sperner]], in relation with proofs of [[invariance of domain]]. Sperner colorings have been used for effective computation of [[fixed point (mathematics)|fixed point]]s and in [[root-finding algorithm]]s, and are applied in [[fair division]] (cake cutting) algorithms. It is now believed to be an intractable computational problem to find a Brouwer fixed point or equivalently a Sperner coloring even in the plane, in the general case. The problem is [[PPAD (complexity) | PPAD-complete]], a complexity class invented by [[Christos Papadimitriou]].
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| According to the Soviet ''Mathematical Encyclopaedia'' (ed. [[I.M. Vinogradov]]), a related 1929 theorem (of [[Bronisław Knaster|Knaster]], [[Karol Borsuk|Borsuk]] and [[Stefan Mazurkiewicz|Mazurkiewicz]]) has also become known as the ''Sperner lemma'' – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the [[Knaster–Kuratowski–Mazurkiewicz lemma]].
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| ==Statement==
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| === One-dimensional case ===
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| [[Image:Sperner1d.svg|thumb|20px|right|]]
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| In one dimension, Sperner's Lemma can be regarded as a discrete version of the [[Intermediate Value Theorem]]. In this case, it essentially says that if a discrete [[Function (mathematics)|function]] takes only the values 0 and 1, begins at the value 0 and ends at the value 1, then it must switch values an odd number of times.
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| === Two-dimensional case ===
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| [[Image:Sperner2d.svg||250px|thumb|left|Two dimensional case]]
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| The two-dimensional case is the one referred to most frequently. It is stated as follows:
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| Given a [[triangle]] ABC, and a triangulation ''T'' of the triangle. The set ''S'' of vertices of ''T'' is colored with three colors in such a way that
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| # A, B and C are colored 1, 2 and 3 respectively
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| # Each vertex on an edge of ABC is to be colored only with one of the two colors of the ends of its edge. For example, each vertex on AC must have a color either 1 or 3.
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| Then there exists a triangle from ''T'', whose vertices are colored with the three different colors. More precisely, there must be an odd number of such triangles.
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| === Multidimensional case ===
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| In the general case the lemma refers to a ''n''-dimensional simplex
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| :<math>\mathcal{A}=A_1 A_2 \ldots A_{n+1}.</math>
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| We consider a triangulation ''T'' which is a disjoint division of <math>\mathcal{A}</math> into smaller ''n''-dimensional simplices. Denote the coloring function as ''f'' : ''S'' → {1,2,3,...,''n'',''n''+1}, where ''S'' is again the set of vertices of ''T''. The rules of coloring are:
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| # The vertices of the large simplex are colored with different colors, i. e. ''f''(''A''<sub>''i''</sub>) = ''i'' for 1 ≤ ''i'' ≤ ''n''+1.
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| # Vertices of ''T'' located on any ''k''-dimensional subface
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| ::<math>A_{i_1}A_{i_2}\ldots A_{i_{k+1}}</math>
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| :are colored only with the colors
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| ::<math>i_1,i_2,\ldots,i_{k+1}.</math>
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| Then there exists an odd number of simplices from ''T'', whose vertices are colored with all ''n+1'' colors. In particular, there must be at least one.
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| == Proof ==
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| We shall first address the two-dimensional case. Consider a graph ''G'' built from the triangulation ''T'' as follows:
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| :The vertices of ''G'' are the members of ''T'' plus the area outside the triangle. Two vertices are connected with an edge if their corresponding areas share a common border with one endpoint colored 1 and the other colored 2.
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| Note that on the interval AB there is an odd number of borders colored 1-2 (simply because A is colored 1, B is colored 2; and as we move along AB, there must be an odd number of color changes in order to get different colors at the beginning and at the end). Therefore the vertex of ''G'' corresponding to the outer area has an odd degree. But it is known (the [[handshaking lemma]]) that in a finite graph there is an even number of vertices with odd degree. Therefore the remaining graph, excluding the outer area, has an odd number of vertices with odd degree corresponding to members of ''T''.
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| It can be easily seen that the only possible degree of a triangle from ''T'' is 0, 1 or 2, and that the degree 1 corresponds to a triangle colored with the three colors 1, 2 and 3.
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| Thus we have obtained a slightly stronger conclusion, which says that in a triangulation ''T'' there is an odd number (and at least one) of full-colored triangles.
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| A multidimensional case can be proved by induction on the dimension of a simplex. We apply the same reasoning, as in the 2-dimensional case, to conclude that in a ''n''-dimensional triangulation there is an odd number of full-colored simplices.
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| == Generalizations ==
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| Sperner's lemma has been generalized to colorings of [[polytope]]s with ''n'' vertices.. In that case, there are at least ''n-k'' fully labeled simplices, where ''k'' is the dimension of the polytope and "fully labeled" indicates that every label on the simplex has a different color. For example, if a polygon with ''n'' vertices is triangulated and colored according to the Sperner criterion, then there are at least ''n-2'' fully labeled triangles. The general statement was conjectured by [[Krassimir Atanassov|Atanassov]] in 1996, who proved it for the case ''k=2''.<ref>{{cite journal|author=K. T. Atanassov|year=1996|title=On Sperner's Lemma | journal=Stud. Sci. Math. Hungarica | volume=32 | pages=71–74}}</ref> The proof of the general case was first given by de Loera, Peterson, and Su in 2002.<ref>{{cite journal|doi=10.1006/jcta.2002.3274|author=Jesus de Loera, Elisha Peterson, and Francis Su |year=2002 |title=A polytopal generalization of Sperner's Lemma | journal=Journal of Combinatorial Theory Series A | volume=100|issue=1 | pages=1–26}}</ref>
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| == Applications ==
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| Sperner colorings have been used for effective computation of [[fixed point (mathematics)|fixed point]]s. A Sperner coloring can be constructed such that fully labeled simplices correspond to fixed points of a given function. By making a triangulation smaller and smaller, one can show that the limit of the fully labeled simplices is exactly the fixed point. Hence, the technique provides a way to approximate fixed points.
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| For this reason, Sperner's lemma can also be used in [[root-finding algorithm]]s and [[fair division]] algorithms. For instance, it can be used to find an [[envy-free]] partition of the rooms and rent in a shared apartment.<ref>{{citation
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| | last = Su | first = Francis Edward
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| | doi = 10.2307/2589747
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| | issue = 10
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| | journal = The American Mathematical Monthly
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| | mr = 1732499
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| | pages = 930–942
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| | title = Rental harmony: Sperner's lemma in fair division
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| | url = http://www.math.hmc.edu/~su/papers.dir/rent.pdf
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| | volume = 106
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| | year = 1999}}. See also [http://agtb.wordpress.com/2012/08/15/fair-division-and-the-whining-philosophers-problem/ Fair division and the whining philosophers problem], Ariel Procaccia, 2012.</ref>
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| Sperner's lemma is one of the key ingredients of the proof of [[Monsky's theorem]], that a square cannot be cut into an odd number of [[equidissection|equal-area triangles]].<ref>{{citation
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| | last1 = Aigner | first1 = Martin | author1-link = Martin Aigner
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| | last2 = Ziegler | first2 = Günter M. | author2-link = Günter M. Ziegler
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| | contribution = One square and an odd number of triangles
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| | doi = 10.1007/978-3-642-00856-6_20
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| | edition = 4th
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| | location = Berlin
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| | pages = 131–138
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| | publisher = Springer-Verlag
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| | title = Proofs from The Book
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| | year = 2010}}</ref>
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| Fifty years after first publishing it, Sperner presented a survey on the development, influence and applications of his combinatorial lemma.<ref>{{cite journal|author=E. Sperner|year=1980|title=Fifty years of further development of a combinatorial lemma | journal=Numerical solution of highly nonlinear problems (Sympos. Fixed Point Algorithms and Complementarity Problems, Univ. Southampton, Southampton, 1979) | volume= | pages=Part A, p.183–197, Part B, p.199–214}}</ref>
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| == See also ==
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| * [[Brouwer fixed point theorem]]
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| * [[Borsuk–Ulam theorem]]
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| * [[Tucker's lemma]]
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| * [[Topological combinatorics]]
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| == References ==
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| <references/>
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| ==External links==
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/SpernerLemma.shtml Proof of Sperner's Lemma] at [[cut-the-knot]]
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| {{DEFAULTSORT:Sperner's Lemma}}
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| [[Category:Combinatorics]]
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Topology]]
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| [[Category:Lemmas]]
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| [[Category:Articles containing proofs]]
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| [[Category:Fair division]]
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