Sperner's lemma

In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which is equivalent to it.

Sperner's lemma states that every Sperner coloring (described below) of a 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 points and in root-finding algorithms, 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-complete, a complexity class invented by Christos Papadimitriou.

According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had 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.

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.

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