Let $V$ be a set of positive integers whose natural density is 1. Is it necessarily true that $V$ contains an infinite arithmetic progression?—i.e., that there are non-negative integers $a,b,\nu$ with $0\leq b\leq a-1$ so that: $$\left\{ an+b:n\geq\nu\right\} \subseteq V$$

In addition to an answer, any references on the matter would be most appreciated.

here. This example arises in a different context [finding arbitrarily long strings of consecutive composite numbers] and/but satisfies the criterion here, too. $\endgroup$ – Benjamin Dickman Jun 9 at 18:45