Applications
Sperner colorings have been used for effective computation of fixed points. A Sperner coloring can be constructed such that fully labeled simplices correspond to fixed points of a given function. By making a triangulation smaller and smaller, one can show that the limit of the fully labeled simplices is exactly the fixed point. Hence, the technique provides a way to approximate fixed points.
For this reason, Sperner's lemma can also be used in root-finding algorithms and fair division algorithms. For instance, it can be used to find an envy-free partition of the rooms and rent in a shared apartment.
Sperner's lemma is one of the key ingredients of the proof of Monsky's theorem, that a square cannot be cut into an odd number of equal-area triangles.
E. Sperner has presented the development, influence and applications of his combinatorial lemma in
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