Monochromatic triangles in two-colored plane

Abstract

We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a translated and rotated copy of $T$ contained in $C$ or in $O$. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration $T$ which is a vertex set of a non-equilateral triangle there is a copy of $T$ contained in the interior of one of the two partition classes. Furthermore, we give the characterization of these "polygonal" partitions that avoid copies of a given equilateral triple. These results support a conjecture of Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, which states that every two-coloring of the plane contains a monochromatic copy of any nonequilateral triple of points; on the other hand, we disprove a stronger conjecture by the same authors, by providing non-trivial examples of two-colorings that avoid a given equilateral triple.