Erdős Problem #199

If $A\subset \mathbb{R}$ does not contain a 3-term arithmetic progression then must $\mathbb{R}\backslash A$ contain an infinite arithmetic progression?

#199: [Er75b][ErGr79][ErGr80]

arithmetic progressions

The answer is no, as shown by Baumgartner [Ba75] (whose construction uses the axiom of choice to provide a basis for $\mathbb{R}$ over $\mathbb{Q}$).

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