# Sperner Lemma Applications

I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of this lemma, namely, Monsky's theorem, which states that it is impossible to dissect a square into an odd number of triangles having equal areas.

Browsing through a few questions this evening I found two references to Sperner's lemma with respect to totally different applications. I searched the site, and didn't found a question which asks about other applications of Sperner's lemma, so I thought I'll ask the question myself.

What other applications of Sperner's lemma are there?

(I made the question community wiki.)

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Francis Su wrote a paper called Rental Harmony: Sperner's Lemma in Fair Division that, as the name indicates, uses Sperner's lemma to solve some fair division problems. It the 2001 Merten Hasse award winning paper, and as such can be found free of charge here

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Ron Aharoni and Penny Haxell have applied Sperner's Lemma to simplicial complexes associated with families of hypergraphs to prove various results on independent transversals. In particular, they obtain a topological proof of Hall's theorem. Here's a paper, and an overview on Gil Kalai's blog.

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