Problem Statement
If $A\subset \mathbb{R}$ does not contain a 3-term arithmetic progression then must $\mathbb{R}\backslash A$ contain an infinite arithmetic progression?
Categories:
Arithmetic Progressions
Progress
The answer is no, as shown by Baumgartner [Ba75] (whose construction uses the axiom of choice to provide a basis for $\mathbb{R}$ over $\mathbb{Q}$).Source: erdosproblems.com/199 | Last verified: January 14, 2026