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If $A\subseteq \mathbb{N}$ is a multiset of integers such that \[\lvert A\cap \{1,\ldots,N\}\rvert\gg N\] for all $N$ then must $A$ be subcomplete? That is, must \[P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}\] contain an infinite arithmetic progression?
A problem of Folkman. Folkman [Fo66] showed that this is true if \[\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1+\epsilon}\] for some $\epsilon>0$ and all $N$.

The original question was answered by Szemerédi and Vu [SzVu06] (who proved that the answer is yes).

This is best possible, since Folkman [Fo66] showed that for all $\epsilon>0$ there exists a multiset $A$ with \[\lvert A\cap \{1,\ldots,N\}\rvert\ll N^{1+\epsilon}\] for all $N$, such that $A$ is not subcomplete.

Additional thanks to: Zachary Hunter