Any 2-coloring of the plane contains monochromatic unit rhombuses

TL;DR

This paper proves that in any 2-coloring of the plane, there must exist four same-colored points forming a rhombus with sides of length one and diagonals of non-unit length.

Contribution

It resolves a previously open question by demonstrating the unavoidable presence of such monochromatic rhombuses in any 2-coloring of the plane.

Findings

Any 2-coloring of the plane contains a monochromatic unit rhombus.

The result answers a question posed by Axenovich, Liu, and the second author.

The proof confirms the existence of specific geometric configurations in 2-colorings.

Abstract

In this note, we prove that any 2-coloring of the plane contains 4 points of the same color forming a rhombus with unit sides and non-unit diagonals, answering a question of Axenovich, Liu, and the second author.