Monochromatic triangles in two-colored plane

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 nonequilateral 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 Erdős, 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.