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Hey,

I'm writhing my bachelor thesis in mathematics and I'm stuck at finding a proof for Sperner's lemma using Brouwer's fixed point theorem. I need it to prove the equivalence of the two theorems.

I would appreciate any help in finding a proof.

Thanks in advance!

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 1. Take the map which projects the vertex of color $n$ to the opposite side. 2. Extend this map linearly to all simplices. 3. Think. (I vote to close.) – Anton Petrunin Jul 29 at 12:42
There are many in the literature. – Fernando Muro Jul 29 at 12:59
Thanks for your replies. A common proof only shows the existence of at least one completely colored simplex, not that there must be an odd number. Anyone know a specific book that covers this? – Isak Kupersmidt Jul 29 at 13:52
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 Maybe try asking this on math stackexchange and you will get a more detailed answer. – Benjamin Steinberg Jul 29 at 15:31
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 If you ask on Math Stackexchange, you should probably include in the question that you want the "odd number" strengthening of Sperner's Lemma, not just the "at least one" version. In my experience, it is the latter that people usually mean when they say "Sperner's Lemma" (especially in connection with Brouwer's theorem, since only the latter version is needed to deduce Brouwer's theorem). – Andreas Blass Jul 29 at 22:33
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closed as too localized by S. Sra, Anton Petrunin, Fernando Muro, Steven Landsburg, Henry Cohn Jul 29 at 14:14

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