Was reading sperners lemma from this
http://www.math.hmc.edu/funfacts/ffiles/20001.4.shtml
Couldn't understand certain things
How to mark internal vertices?
I could have mark some other number for the internal vertices and get another result!
2026-03-30 06:09:22.1774850962
Sperners lemma how to mark internal vertices
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In Sperner's Lemma, the only restrictions on the vertex labeling are for boundary vertices. Vertices which are in the interior of the triangle (or the simplex, in general) can be labelled arbitrarily.
Indeed, this is the real power of the theorem. As you mention, it is easy to imagine that some labeling of the interior vertices would falsify the conclusion. However, it turns out that any labeling of the interior vertices allows you to find a fully-labeled simplex (a "baby (123) triangle").