Any two-coloring of the plane contains monochromatic 3-term arithmetic progressions

Abstract

A conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus states that, with the exception of equilateral triangles, any two-coloring of the plane will have a monochromatic congruent copy of every three-point configuration. This conjecture is known only for special classes of configurations. In this manuscript, we confirm one of the most natural open cases; that is, every two-coloring of the plane admits a monochromatic congruent copy of any $3$-term arithmetic progression.

13 pages, revised based on referee comments

References in corpus (1)

• Any two-coloring of the plane contains monochromatic 3-term arithmetic progressions

Cited by in corpus (4)

• Any two-coloring of the plane contains monochromatic 3-term arithmetic progressions
• Progressions in Euclidean Ramsey theory
• Avoiding short progressions in Euclidean Ramsey theory
• Monochromatic triangles in the max-norm plane