  A conjecture of Leader, Russell and Walters in Euclidean Ramsey theory says
that a finite set is Ramsey if and only if it is congruent to a subset of a set
whose symmetry group acts transitively. As they have shown the ``if" direction
of their conjecture follows if all finite groups have a Hales--Jewett type
property. In this paper, we show that this property is satisfied in the case of
finite solvable groups. Our result can be used to recover the work of
K\v{r}\'i\v{z} in Euclidean Ramsey theory.
