A van der Waerden variant (Q1288044)

Summary: The classical van der Waerden theorem says that for 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.