Problem #WSP-000187

Problem

Banksy used three colours to colour the entire plane: red, green and gold. Show that there are two points, distance one apart, which are the same colour.
The same is true if he used $4$ colours, including grey for example, but that is significantly more difficult to show.

This is part of a more general problem of colouring the entire plane with some number of colours and trying to avoid segments of length $1$ with both ends of the same colour. The question is, how many colours do we need? We know $7$ is enough and that $3$ is not enough. The general problem is known as Hadwiger–Nelson problem and was first formulated in 1950, with those bounds found almost immediately. The next piece of progress was only made in 2018, when Aubrey de Grey found a geometrical figure with $1581$ points (we used only $7$ in the above proof) that shows it is not possible to only use $4$ colours. That was later refined to $510$ points. The general answer of whether it is $5$, $6$ or $7$ colours is still unknown.