Colour profile of a finite block ($(c(b_1),\dots ,c(b_N))$)

The colour profile of a length-$N$ block $b_1,\dots ,b_N$ under a finite colouring $c$ is the ordered tuple $(c(b_1),\dots ,c(b_N))$. There are at most $r^N$ profiles when $c$ uses at most $r$ colours. Equally profiled blocks reproduce every colour configuration at the same relative positions; this converts a colouring of points into a finite colouring of blocks in the colour-focusing proof of Van der Waerden theorem.