Line segment is a Euclidean Ramsey set

The two endpoints of a nonzero line segment form a Euclidean Ramsey set. The finite configuration consists of those two points, rather than every point of the continuous line segment.

For a finite colouring with $q$ colours and a desired distance $d>0$, take the $q+1$ vertices of a regular simplex of side length $d$ in ${\mathbb{R}}^q$. The pigeonhole principle gives two vertices of the same colour. Every pair of distinct vertices has distance $d$, so this pair is an isometric copy of the prescribed endpoint pair. This also gives an explicit sufficient ambient dimension for every number of colours.