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{\bf Kevin J. Compton}
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{\bf A van der Waerden Variant}
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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 {\it 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.



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