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 $d_\ell\to \infty$?