Sperner's Lemma states that for a triangulation of a simplex whose vertices are colored according to a specific boundary rule, at least one of the small triangulated cells must have all of its own vertices colored with the full set of colors. Named for Emanuel Sperner, it is a purely combinatorial statement that nonetheless supplies one of the simplest known proofs of the Brouwer Fixed-Point Theorem, since the fully colored cell it guarantees can be used to locate a fixed point as the triangulation is refined.
Facts
StatementEvery Sperner coloring of a triangulation of an n-dimensional simplex contains a cell whose vertices all have different colors. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Source Sperner's lemma (Wikipedia)
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1. Sperner's lemma (Wikipedia)
Statement
every Sperner coloring of a triangulation of an n-dimensional simplex contains a cell whose vertices all have different colors.
Proved By: Emanuel Sperner, Lead paragraph
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is
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