Consider all the points in the plane and assign each one a color so that any two points one unit apart have different colors.
An equilateral triangle with side length 1 shows that two colors are not enough to color every point in the plane. Nor, in fact, are three, as the Moser graph below shows.
Furthermore, seven colors are enough. To see this, we can tile the plane with hexagons whose diameter is slightly less than one unit.
This raises a question: what is the minimum number of colors required? Four, five, six, or seven? The answer is not straightforward. It was not until April 2018 that Aubrey de Grey, a British biologist specializing in aging but also an amateur mathematician, proposed a graph made up of line segments of length 1 that requires five colors! This graph has more than twenty thousand vertices, but it has since been simplified, and we now have an example with just over eight hundred vertices. So, does it take five, six, or seven colors to color the plane?
