Sperner's lemma

Sperner's lemma states that a suitably labeled triangulation of a simplex contains a small simplex whose vertices carry every label. It guarantees such a simplex and, in its parity form, an odd number of them.

Simplex: A simplex is the convex hull of a finite set of affinely independent points. The lemma’s domain is a simplex, whose boundary faces make the labeling rule work.
Barycentric subdivision: Barycentric subdivision divides a simplicial complex into smaller simplices using the barycenters of its faces. Repeated subdivision produces the fine triangulations on which the lemma can be applied.
Brouwer fixed-point theorem: Brouwer's fixed-point theorem states that every continuous map from a compact convex set to itself has a fixed point. Sperner labelings yield approximate fixed points on increasingly fine triangulations.
Kakutani fixed-point theorem: Kakutani's fixed-point theorem guarantees a fixed point for an upper hemicontinuous correspondence with nonempty compact convex values on a compact convex domain. It extends fixed-point existence to set-valued maps, whereas Sperner’s statement is combinatorial.
Triangulation: A triangulation partitions a geometric space into simplices that meet along shared faces. The lemma examines the small simplices in a triangulated domain.
Simplicial complex: A simplicial complex is a collection of simplices joined along shared faces. A triangulation of the domain is organized as a simplicial complex.
Nash equilibrium: A Nash equilibrium is a strategy profile in which no player can improve their payoff by changing strategy alone. A discretized proof uses Sperner’s lemma to establish equilibrium existence.
Knaster–Kuratowski–Mazurkiewicz lemma: The Knaster–Kuratowski–Mazurkiewicz lemma gives an intersection condition for finite covers of a simplex. It is a closely related combinatorial-topological lemma with equivalent forms.
Sperner coloring: A Sperner coloring labels vertices of a triangulation so each boundary face uses only labels belonging to that face. This boundary constraint is the hypothesis that forces a fully labeled simplex.
Incidence counting: Incidence counting counts relationships between objects, such as faces shared by adjacent simplices. The proof tracks how often labeled faces belong to neighboring simplices.

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Monsky's theoremRelated: The proof uses a colored-triangle principle closely related to Sperner's lemma.