I have the following statement which is claimed to be a version of Van der Waerden's theorem:
For any finite partition of $\mathbb{N}$, one of the cells contains affine images of every finite set.
This statement is taken from the book Elemental methods in ergodic ramsey theory.
The affine version of the Van der Waerden's theorem I had understood before was:
For any finite partition of $\mathbb{N}$ and any finite subset of $\mathbb{N}$, one of the cells contains an affine image of it.
My first question is, isn't this what the book actually means, because it's not quite clear to me how the two statements are linked? My second question is, how to show that the above affine version is equivalent to the statement obtained by replacing $\mathbb{N}$ by $\mathbb{Z}$?
Thanks for your time.