Sperner's Lemma

Sperner's Lemma --- Лемма Шпернера.

Lemma. Let [math]\displaystyle{ T }[/math] be a triangulation of [math]\displaystyle{ \Delta_{n} }[/math] and let [math]\displaystyle{ \chi }[/math] be a coloring of the points of [math]\displaystyle{ T }[/math] by [math]\displaystyle{ n+1 }[/math] colors, which satisfies the following conditions:

1. Each vertex of [math]\displaystyle{ \Delta_{n} }[/math] is colored by a different color.

2. The points of [math]\displaystyle{ T }[/math] on a face [math]\displaystyle{ \tau }[/math] of [math]\displaystyle{ \Delta_{n} }[/math] are colored by the vertices of [math]\displaystyle{ \tau }[/math].

Then there exists a simplex in the triangulation, whose vertices receive all [math]\displaystyle{ n+1 }[/math] colors.