Let $W(r, k)$ be a van der Waerden number, such that the interval $[1, W(r, k)]$ contains an arithmetic progression (AP) of $k$ terms, (k > 1), where the integers in the AP all have the same (monochromatic) coloring.
Let $n$ be the positive integer exponent for which $r^{n}$ divides $W(r, k)$ while $r^{n + 1}$ does not divide $W(r, k)$.
Let $$ b_{n}, b_{n - 1}, \ldots, b_{0} \in [0, r - 1], $$ all be integers, with $$ 1 \leq b_{n} < r, $$ such that \begin{equation} W(r, k) = b_{n}r^{n} + b_{n - 1}r^{n - 1} + \ldots + b_{0}. \end{equation} QUESTION: Does anyone here claim that the following inequalities below are false when the arithmetic is done in base ten? \begin{equation} r^{n} \leq b_{n}r^{n} + b_{n - 1}r^{n - 1} + \ldots + b_{0} < r^{n + 1}. \end{equation} That is to say, since $$ W(r, k) = b_{n}r^{n} + b_{n - 1}r^{n - 1} + \ldots + b_{0}, $$ we have that \begin{equation} W(r, k) < r^{n + 1}. \end{equation} If anyone here claims these conclusions are false, then I am very much interested in seeing some helpful and valid counterarguments.
Here are three particular examples of what I mean when the number of colorings are respectively, (r = 3, r = 4, r = 2), and where (W(3, 3) = 27, W(4, 3) = 76, W(2, 6) = 1132). \begin{eqnarray} 3^{3}&\leq&W(3, 3) = 3^{3} < 3^{4},\nonumber\\ 4^{3}&\leq&W(4, 3) = 4^{3} + 3(4^{1}) < 4^{4},\nonumber\\ 2^{10}&\leq&W(2, 6) = 2^{10} + 2^{6} + 2^{4} + 2^{2} < 2^{11}. \end{eqnarray}