Sperner's lemma

One-dimensional case example

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 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.

Two-dimensional case example

The two-dimensional case is the one referred to most frequently. It is stated as follows:

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

1. A, B, and C are colored 1, 2, and 3 respectively
2. 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.

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.

In the general case the lemma refers to a n-dimensional simplex

${\displaystyle {\mathcal {A}}=A_{1}A_{2}\ldots A_{n+1}.}$

We consider a triangulation T which is a disjoint division of ${\displaystyle {\mathcal {A}}}$ 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:

1. The vertices of the large simplex are colored with different colors, i. e. f(A_i) = i for 1 ≤ i ≤ n+1.
2. Vertices of T located on any k-dimensional subface of the large simplex

${\displaystyle A_{i_{1}}A_{i_{2}}\ldots A_{i_{k+1}}}$

are colored only with the colors

${\displaystyle i_{1},i_{2},\ldots ,i_{k+1}.}$

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.

A simple two-dimensional triangulation of the example figure, colored and named in accordance with the assumptions of Sperner's Lemma

The graph derived from the example figure

Here is an elaboration of the proof given previously, for a reader new to graph theory.

This diagram numbers the colors of the vertices of the example given previously. The small triangles whose vertices all have different numbers are shaded in the graph. Each small triangle becomes a node in the new graph derived from the triangulation. The small letters identify the areas, eight inside the figure, and area i designates the space outside of it.

As described previously, those nodes that share an edge whose endpoints are numbered 1 and 2 are joined in the derived graph. For example, node d shares an edge with the outer area i, and its vertices all have different numbers, so it is also shaded. Node b is not shaded because two vertices have the same number, but it is joined to the outer area.

One could add a new full-numbered triangle, say by inserting a node numbered 3 into the edge between 1 and 1 of node a, and joining that node to the other vertex of a. Doing so would have to create a pair of new nodes, like the situation with nodes f and g.

Suppose that each vertex of the triangulation may be labeled with multiple colors, so that the coloring function is f : S → 2^[n+1].

For every sub-simplex, the set of labelings on its vertices is a set-family over the set of colors [n+1]. This set-family can be seen as a hypergraph.

If, for every vertex v on a face of the simplex, the colors in f(v) are a subset of the set of colors on the face endpoints, then there exists a sub-simplex with a balanced labeling - a labeling in which the corresponding hypergraph admits a perfect fractional matching. To illustrate, here are some balanced labeling examples for n=2:

• ({1}, {2}, {3}) - balanced by the weights (1, 1, 1).
• ({1,2}, {2,3}, {3,1}) - balanced by the weights (1/2, 1/2, 1/2).
• ({1,2}, {2,3}, {1}) - balanced by the weights (0, 1, 1).

This was proved by Shapley in 1973.[2] It is a combinatorial analogue of the KKMS lemma.

Suppose that, instead of an ${\displaystyle n-1}$-dimensional simplex, we have a ${\displaystyle d}$-dimensional polytope with ${\displaystyle n}$ vertices.

Then there are at least ${\displaystyle n-d}$ fully labeled simplices, where "fully labeled" indicates that every label on the simplex has a different color. For example, if a (two-dimensional) polygon with n vertices is triangulated and colored according to the Sperner criterion, then there are at least ${\displaystyle n-2}$ fully labeled triangles.

The general statement was conjectured by Atanassov in 1996, who proved it for the case ${\displaystyle d=2}$.[3] The proof of the general case was first given by de Loera, Peterson, and Su in 2002.[4]

Suppose that, instead of a single labeling, we have ${\displaystyle n}$ different Sperner labelings.

We consider pairs (simplex,permutation) such that, the label of each vertex of the simplex is chosen from a different labeling (so for each simplex, there are ${\displaystyle n!}$ different pairs).

Then there are at least ${\displaystyle n!}$ fully labeled pairs. This was proved by Ravindra Bapat.[5]

Another way to state this lemma is as follows. Suppose there are ${\displaystyle n}$ people, each of whom produces a different Sperner labeling of the same triangulation. Then, there exists a simplex, and a matching of the people to its vertices, such that each vertex is labeled by its owner differently (one person labels its vertex by 1, another person labels its vertex by 2, etc.). Moreover, there are at least ${\displaystyle n!}$ such matchings. This can be used to find an envy-free cake-cutting with connected pieces.

Suppose that, instead of a single labeling, we have ${\displaystyle m}$ different Sperner labelings. Then:[6]^{:Thm 2.1}

1. For any positive integers ${\displaystyle k_{1},\ldots ,k_{m}}$whose sum is ${\displaystyle m+n-1}$, there exists a baby-simplex on which, for every ${\displaystyle i\in \{1,\ldots ,m\}}$, labeling number ${\displaystyle i}$ uses at least ${\displaystyle k_{i}}$ (out of ${\displaystyle n}$) distinct labels. Moreover, each label is used by at least one (out of ${\displaystyle m}$) labeling.
2. For any positive integers ${\displaystyle l_{1},\ldots ,l_{n}}$whose sum is ${\displaystyle m+n-1}$, there exists a baby-simplex on which, for every ${\displaystyle j\in \{1,\ldots ,n\}}$, the label ${\displaystyle j}$ is used by at least ${\displaystyle l_{j}}$ (out of ${\displaystyle m}$) different labelings.

Both versions reduce to Sperner's lemma when ${\displaystyle m=1}$, or when all ${\displaystyle m}$ labelings are identical.

See [7] for similar generalizations.

Suppose a triangle is triangulated and labeled with {1,2,3}. Consider the cyclic sequence of labels on the boundary of the triangle. Define the degree of the labeling as the difference between the number of switches from 1 to 2, and the number of switches from 2 to 1. See examples in the table at the right. Note that the degree is the same if we count switches from 2 to 3 minus 3 to 2, or from 3 to 1 minus 1 to 3.

Musin proved that the number of fully labeled triangles is at least the degree of the labeling.[8] In particular, if the degree is nonzero, then there exists at least one fully labeled triangle.

If a labeling satisfies the Sperner condition, then its degree is exactly 1: there are 1-2 and 2-1 switches only in the side between vertices 1 and 2, and the number of 1-2 switches must be one more than the number of 2-1 switches (when walking from vertex 1 to vertex 2). Therefore, the original Sperner lemma follows from Musin's theorem.

There is a similar lemma about finite and infinite trees and cycles.[9]

A variant of Sperner's lemma on a cube (instead of a simplex) was proved by Harold W. Kuhn.[10] It is related to the Poincaré–Miranda theorem.[11]