Problem #847: Let $A\subset \mathbb{N}$ be an infinite set for which...

Let $A\subset \mathbb{N}$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there is a subset of size at least $\epsilon n$ which contains no three-term arithmetic progression.

Is it true that $A$ is the union of a finite number of sets which contain no three-term arithmetic progression?