On monochromatic triangles (Q797151)

The author first gives a particularly simple proof of J. B. Kelly's theorem: If A and B are two finite sets in \(R^ d\) (\(d\geq 2)\) with every open segment joining two points in the same set containing a point of the other, then all the points of A and B lie on a line. Then he generalizes to: If A and B are finite disjoint sets and A contains \(2d+1\) points in a general position then there is a d-simplex with vertices in one of the sets and no point of either set in its interior. In another theorem (in \(R^ 2)\) conditions are given that guarantee the existence of a triangle with vertices in one of the sets and no other point (from either set) on its boundary.