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 of the colors on the boundaries of the strips.

We decided to verify these two conjectures for special classes of colorings. First, we proved that if one of the color classes of the coloring is an open set, then the coloring contains a monochromatic copy of any triple of points. Apart from that, we considered colorings where the boundary between the two color classes is a locally-finite union of straight line segments. We showed that these colorings contain all non-equilateral triples; however, this class of colorings contains counterexamples to the second conjecture. We are able to fully characterize these counterexamples, which may be regarded as a generalization of the coloring by parallel strips.