Erdős Problem #847

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? A negative answer was given by Reiher, Rödl, and Sales [RRS24], who proved that, for any \(0<\mu<1/2\), there exists \(A\subseteq \mathbb{N}\) such that every finite colouring of \(A\) contains a three-term arithmetic progression, and yet every subset of \(A\) of size \(n\) contains a subset of size \(\geq \mu n\) without a three-term arithmetic progression.