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Claim. Y.-G. Chen, On subset sums of a fixed set, Acta Arith. 106 (2003), no. 3, 207--211. Its Theorem 1: there is an absolute constant $C>1$ such that every strictly increasing sequence $A$ of positive integers with $\lvert A(n)\rvert>C\sqrt n$ for all $n>C^2$, where $A(n)$ is the set of elements of $A$ not exceeding $n$, is subcomplete, so the set $P(A)$ of finite subset sums of $A$ contains an infinite arithmetic progression. This answers the question of Problem 344 yes, in the reading in which the hypothesis $\lvert A\cap\{1,\ldots,N\}\rvert\gg N^{1/2}$ carries a sufficiently large implied constant; the problem page records why the constant is needed. The paper precedes both proofs of Szemerédi and Vu, whose J. Amer. Math. Soc. paper records in the closing remark of its Section 9 that Chen proved the same theorem by a different method (see their claim page). The journal's record gives only the year, so the page is dated 2003-01-01 by convention.

Acceptance. Refereed: Acta Arithmetica. The site's curator credits the resolution of Problem 344 to Szemerédi and Vu and does not cite Chen, so no reviewed evidence is listed. Nothing here was checked by this project.