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.)


Consider a random $2$-coloring of the natural numbers. A given infinite arithmetic progression has probability zero of being monochromatic. As there are only countably many infinite arithmetic progressions, with probability one there is no monochromatic infinite arithmetic progression.
More generally, a random $k$-coloring of $\mathbb N$ will almost surely have no infinite arithmetic progression in which only $k-1$ colors occur.
