A van der Waerden Variant

The classical van der Waerden Theorem says that for every every finite set S of natural numbers and every k-coloring of the natural numbers, there is a monochromatic set of the form aS+b for some a > 0 and b ≥ 0. I.e., monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. S has the variant van der Waerden property for k colors if for every k-coloring there is a monochromatic set of the form a(S+ b) for some a > 0 and b ≥ 0. On the positive side it is shown that every two-element set has the variant van der Waerden property for every k. Also, for every finite S and k there is an n such that nS has the variant van der Waerden property for k colors. This extends the classical van der Waerden Theorem. On the negative side it is shown that if S has at least three elements, the variant van der Waerden property fails for a sufficiently large k. The counterexamples to the variant van der Waerden property are constructed by specifying colorings as Thue-Morse sequences. Submitted July 17, 1997; Accepted April 2, 1999. AMS Subject Classification. Primary: 05D10. Secondary: 11B85, 68R15.