A Ramsey-type bound for rectangles

It is proved that for any rectangle T and for any 2-coloring of the points of the 5-dimensional Euclidean space, one can always find a rectangle T' congruent to T, all of whose vertices are of the same color. We also show that for any k-coloring of the k(2) + o(k(2))-dimensional space, there is a monochromatic rectangle congruent to any given rectangle. (C) 1996 John Wiley & Sons, Inc.