  We use ergodic theoretic tools to solve a classical problem in geometric
Ramsey theory. Let E be a measurable subset of R^m, with positive upper
density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large
enough, we can find an isometric copy of rV arbitrarily close to E. This is a
generalization of a theorem of Furstenberg, Katznelson and Weiss showing a
similar property for m=k=2.
