Recall that a syndetic subset of the integers is any $S\subseteq \mathbb{Z}$ with bounded gaps, i.e. there is some $k< \omega$ so that consecutive members of $S$ have distance at most $k$. One way to state Van der Waerden's theorem is that every syndetic set contains arbitrarily long arithmetic progressions (which I will call APs). 


Call an AP of the form $\{a, a+d,...,a+(\ell-1)d\}$ a $d$-AP of length $\ell$. Now fix a syndetic $S\subseteq \mathbb{Z}$. For each $\ell < \omega$, let $d\_\ell$ denote the least $d$ so that $S$ contains a $d$-AP of length $\ell$. My question is about the function $\ell\to d\_\ell$. Is there a syndetic $S\subseteq \mathbb{Z}$ where $d\_\ell\to \infty$?



Arithmetic progressions in Van der Waerden's theorem
The answer is yes, in fact the sequence $S\subset \mathbb{N}$ indexed by [Thue-Morse sequence](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence) works, see [this MO question](https://mathoverflow.net/questions/87923/sequences-without-long-arithmetic-progressions). Note that this sequence is syndetic because it contains one of $\{2n, 2n+1\}$ for every $n$. You can extend to the negatives by reflecting and trivially only lose a factor of two on $\ell$.


