Monochromatic triangles in two-colored plane

Vít Jelínek

Abstract

Consider the following question: for a given configuration T of three points in the plane, is it possible to color the plane with two colors so that neither of the two color classes contains a rotated and translated copy of the configuration T?

If T is the vertex set of an equilateral triangle, it is easy to see that a coloring of the plane into parallel half-open strips whose widths are equal to the height of T contains no monochromatic copy of T. In 1973, Erdos, Graham, Montgomery, Rothschild, Spencer and Straus conjectured that this example is essentially unique; more precisely, they made the following two conjectures:

Conjecture 1: any two-coloring of the plane contains a monochromatic copy of any non-equilateral triple of points.

Conjecture 2: the only coloring that has no monochromatic copy of an equilateral triple of points is the above-described coloring by parallel strips, up to a possible modification