Coloring the Odd Distance Graph

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

The odd-distance graph $\mathcal O$ has vertex set $\mathbb R^2$ and joins distinct points $x,y$ when their Euclidean distance is an odd positive integer. A proper coloring assigns different colors to adjacent points; no measurability restriction is imposed. A measurable finite coloring means that each color class is Lebesgue measurable.

Statement recorded in the supplied source.

Is there no proper coloring $\mathbb R^2\to\{1,\ldots,m\}$ of $\mathcal O$ for any finite $m$, that is, is $\chi(\mathcal O)$ infinite? The accompanying measurable-coloring variant asks the same question with measurable color classes. [S2]

Short English statement

Must every finite coloring of the plane contain two equally colored points at an odd integer distance, even when color classes are arbitrary sets?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3230 (OPG-616). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Coloring the Odd Distance Graph, original node 616, statement and mathematical discussion. https://www.openproblemgarden.org/op/coloring_the_odd_distance_graph.