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.