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Sperner's lemma - mklabgr - 07-28-2026 Sperner's lemma Summary Sperner's lemma is a combinatorial theorem in mathematics that concerns the colorings of triangulations on simplices, stating that any valid Sperner coloring of a triangulated $n$-dimensional simplex must contain an odd number of fully labeled sub-simplices whose vertices all possess distinct colors.  Proved by Emanuel Sperner in 1928, this lemma is widely recognized for its application in algebraic topology, where it serves as a combinatorial equivalent to the Brouwer fixed-point theorem and is used to guarantee the existence of fixed points in continuous functions. ARTICLE