Special cases of Szemeredi's Theorem?

Question

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.)

Answer 1

The Green-Tao theorem comes to mind. It states that the primes contain arithmetic progressions of arbitrary length. If you want to search for other 'non-trivial' special cases, starting with this Erdos conjecture might be a good idea.

There are also really interesting special cases of Szemeredi's Theorem that were proved before and are more accessible such as Roth's Theorem. This theorem states that any subset of integers of positive density contains a $3$ term arithmetic progression. This theorem is particularly interesting because there has been work on improving the density value. A good survey article is here.

Answer 2

I am not sure if this qualifies under your idea 'slightly non-trivial', but van der Waerden's Theorem is a special case of Szemerédi's Theorem as any finite partition of $\mathbb{Z}$ will necessarily have at least one partition class that has positive upper density, hence guaranteeing the existence of arbitrarily long arithmetic progressions in that partition class.

Although the proof (laid out here rather tersely by McCutcheon in $\S~2.1$) of van der Waerden using combinatorial arguments is certainly easier and shorter than a proof of Roth's Theorem (at least using Furstenberg's Ergodic approach, I am personally not familiar with Roth's original treatment), it would certainly constitute either a rather long MSE post or necessitate a very terse exposition.