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.



Is there a set of positive integers of density 1 which contains no infinite arithmetic progression?
Another construction is to let $n \notin V$ if and only if $n$ begins with at least $\sqrt{\log{n}}$ consecutive '9's when written in decimal. This satisfies the stronger property that there is no non-constant polynomial $f : \mathbb{N} \rightarrow \mathbb{N}$ whose image is contained entirely in $V$.


