Sperner's Lemma

In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from 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, in root-finding algorithms, and are applied in fair division (cake cutting) algorithms. Unfortunately 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 Papadimitriou.

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