On Lipschitz Mappings Onto a Square

Recently Preiss [4] proved that every subset of the plane of a positive Lebesgue measure can be mapped onto a square by a Lipschitz map. In this note we give an alternative proof of this result, based on a well-known combinatorial lemma of Erdős and Szekeres. The validity of an appropriate generalization of this lemma into higher dimensions remains as an open problem.