# Extreme direction Sperner for square 0-1 matrix

Let A be an $n \times n$ 0-1 matrix, and suppose that the facets of the polyhedron $P=\{x: A x \leq {\mathbf 1},\ x \leq {\mathbf 1}\}$ are coloured by $n$ colours such that a facet with extreme direction $-e_i$ does not get colour $i$, and every colour appears twice. Can we find in polynomial time a vertex that is incident to facets of every colour (a panchromatic vertex)?