Multidimensional Case
In the general case the lemma refers to a n-dimensional simplex
We consider a triangulation T which is a disjoint division of into smaller n-dimensional simplices. Denote the coloring function as f : S → {1,2,3,...,n,n+1}, where S is again the set of vertices of T. The rules of coloring are:
- The vertices of the large simplex are colored with different colors, i. e. f(Ai) = i for 1 ≤ i ≤ n+1.
- Vertices of T located on any k-dimensional subface
- are colored only with the colors
Then there exists an odd number of simplices from T, whose vertices are colored with all n+1 colors. In particular, there must be at least one.
Read more about this topic: Sperner's Lemma
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