Are there examples of special cases of Szemeredi's theorem which one can give, which are slightly non-trivial?
To clarify, I'm looking for sets of integers where we can show that they contain infinitely many arithmetic progressions of any finite length. When I say "slightly non-trivial", this is a non-well-defined condition which attempts to remove commenters saying things like "the even numbers" or "all infinite arithmetic progressions" - I'm looking for something like a proof for the special case of the square-free integers.
(Final comment: This is less of a question that I particularly need answered but more of an attempt to collate a collection of such "slightly non-trivial" proofs for my and others' appreciation.)