A van der Waerden Variant

Abstract

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 $b\geq 0$. 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 $b\geq 0$. 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.