SCHWEITZER 1979 Problem 7

Let $T$ be a triangulation of an $n$ -dimensional sphere, and to each vertex of $T$ let us assign a nonzero vector of a linear space $V$ . Show that if $T$ has an $n$ -dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist.

L. Lovasz