= Line segment is a Euclidean Ramsey set
The \b[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 <isometry>[isometric copy] of the prescribed endpoint pair. This also gives an explicit sufficient ambient <dimension> for every number of colours.
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