PreprintCC BY 4.0arXiv
A 21-Coloring of the Plane Without Monochromatic Unit-Area Rectangles
Abstract
Erdős and Graham asked whether every finite coloring of the plane must contain a monochromatic rectangle of any prescribed area. Kovač gave a negative answer by constructing a $25$-coloring with no monochromatic rectangle of area $1$. We reduce the number of colors to $21$ by replacing the square cells in his construction with regular hexagons.
Related
- Two-coloring cubic graphs with small monochromatic components, but without singletons
- Plane separating continua inscribe rectangles
- Two-coloring the points of a projective plane
- Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles
- A Moser-spindle-free 5-chromatic unit distance graph on 2131 vertices in the plane
BibTeX
@misc{hu202621coloring,
title = {{A 21-Coloring of the Plane Without Monochromatic Unit-Area Rectangles}},
author = {Gary Hu and Yumo Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00583},
eprint = {2609.00583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00583},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A 21-Coloring of the Plane Without Monochromatic Unit-Area Rectangles
AU - Gary Hu
AU - Yumo Liu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00583
UR - https://arxiv.org/abs/2609.00583
AB - Erdős and Graham asked whether every finite coloring of the plane must contain a monochromatic rectangle of any prescribed area. Kovač gave a negative answer by constructing a $25$-coloring with no monochromatic rectangle of area $1$. We reduce the number of colors to $21$ by replacing the square cells in his construction with regular hexagons.
ER -