Erdős Problem #199 — Three-Term Progression-Free Sets and Infinite Progressions

The assertion is false: it is not true that every set $A\subseteq\mathbb{R}$ containing no three-term arithmetic progression has an infinite arithmetic progression $S\subseteq\mathbb{R}$ with $S\subseteq\mathbb{R}\setminus A$. Equivalently, there exists a three-term-progression-free set $A\subseteq\mathbb{R}$ whose complement contains no infinite arithmetic progression.