An example showing that van der Waerden's theorem is not true for infinite arithmetic progressions One of the possible formulations of Van der Waerden's theorem is the following:

If $\mathbb N=A_1\cup \dots\cup A_k$ is a partition of the set $\mathbb N$, then one of the sets $A_1,\dots,A_k$ contains finite arithmetic progressions of arbitrary length.

In the other words, if we color the set of all positive integers by finitely many colors, there must be a monochromatic set containing arbitrarily long finite arithmetic progressions.
I assume that the same result is not true if we require that one of the sets contain an infinite arithmetic progression.  (Otherwise this result would be well-known.)

What is an example showing that this stronger version of the above theorem is not true? (I.e., an example of  a coloring of $\mathbb N$ by finitely many colors with no monochromatic infinite arithmetic progression.)


Color the integers black or white according to whether they have an even or an odd number of digits.
