23.37*. A plane is painted in seven colors. Is it necessarily true that there will be two points of the same color, the distance between which is $1$?
23.37. Let's provide an example of coloring the plane in seven colors such that the distance between any two points of the same color is not equal to 1. We will divide the plane into equal hexagons with side length $a$ and color them as shown in Fig. 23.20 (points belonging to two or three hexagons can be colored in any of the colors of these hexagons). The greatest distance between points of the same color within one hexagon does not exceed $2a$, while the smallest distance between points of the same color in different hexagons is not less than the length of segment $AB$ (see Fig. 23.20). It is clear that $AB^2 = AC^2 + BC^2 = 4a^2 + 3a^2 = 7a^2 > (2a)^2$. Therefore, if $2a < 1 < \sqrt{7}a$, i.e., $1 / \sqrt{7} < a < 1 / 2$, then the distance between points of the same color cannot be equal to 1.