Does triangulation have to be finite for Sperner's Lemma to Apply?

Question

I'm a little confused about a proof I read for Sperner's Lemma. The context was described as follows: Assume we have a $n$-dimensional simplex that is partitioned into smaller simplices that are either disjoint or share a full face. Sperner's Lemma states that a proper coloring of such a simplical subdivision must contain a simplex whose vertices share no colors in common, and in particular, there must be an odd number of such simplices (mod $n$).

However, I'm a little confused by the proofs I've seen for the 1-dimension and 2-dimension cases. If we have the simplex as a line segment, it doesn't seem obvious to me that if we have a non-finite number of vertices in the interior of the segment that there need be an odd number of line segments that have differently colored endpoints; couldn't there be an infinite number of such segments? Similarly, the proof I saw for the 2d case seems to use the handshake lemma, which only applies to finite graphs.

Does this mean that Sperner's Lemma only applies to finite partitions of a simplex? I'd appreciate some help clarifying this!

Answer 1

Sperner's lemma only applies to finite partition. But it is often applied on a sequence of finer and finer partitions, colored by a single function from the points of the main simplex to the set of colors. Then by Bolzano-Weierstrass we get a point that is a limit of monochromatic sequences, one for each color. So even if the set of vertices is restricted to be finite, we still get a global result by applying it to infinitely many partitions.

Answer 2

In Sperner's Lemma we consider an $n$-dimensional simplex that is partitioned into smaller $n$-dimensional simplices that are either disjoint or share a full face. Let us call it a regular partition. The finiteness of such a partition is an additional requirement - but an essential one.

In fact there are infinite regular partitions. Here is an example:

Consider the $1$-simplex $[-1,1]$ and partition it into $[-1,0]$ and $[\frac{1}{n+1},\frac{1}{n}]$, $n \in \mathbb N$.

But here is the problem: There exist Sperner coloring functions producing infinitely many $1$-simplices whose vertices are colored by $2$ colors, and it does not make sense to say that there is an odd number of such simplices.