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Claim. Theorem 1.3 of J. Folkman, On the representation of integers as sums of distinct terms from a fixed sequence, Canad. J. Math. 18 (1966), 643--655: a strictly increasing sequence $A=\{a_1<a_2<\cdots\}$ of positive integers with $a_n<Mn^{1+a}$ for all $n$, for some constants $M$ and $0<a<1$, is subcomplete, so the set $P(A)$ of finite subset sums of $A$ contains an infinite arithmetic progression. A set $A$ with $\lvert A\cap\{1,\ldots,N\}\rvert\ge cN^{1/2+\epsilon}$ for all $N$ and some $c>0$, $0<\epsilon<1/2$, satisfies the hypothesis: exactly $n$ elements of $A$ lie in $\{1,\ldots,a_n\}$, so $a_n\le(n/c)^{2/(1+2\epsilon)}$, which is $a_n<Mn^{1+a}$ with $a=(1-2\epsilon)/(1+2\epsilon)$; a larger $\epsilon$ implies the hypothesis for a smaller one. The theorem therefore answers the question of Problem 344 yes for every set whose counting function is $\gg N^{1/2+\epsilon}$ for some $\epsilon>0$, which is the result the site's commentary credits to Folkman. The paper deduces its Theorems 1.1 and 1.2 from it, completeness of such sequences under the necessary residue condition, the second of which proves a conjecture of Erdős in full; the library card folkman_1966_representation_integers_as_sums_distinct_terms digests the paper. Szemerédi and Vu later removed the $\epsilon$ (their claim page), as did Chen (his claim page). The journal's record gives only the year, so the page is dated 1966-01-01 by convention.

Covers. The sets $A$ with $\lvert A\cap{1,\ldots,N}\rvert\gg N^{1/2+\epsilon}$ for some $\epsilon>0$: for them the answer is yes. Nothing at the exponent $1/2$ itself, which is the problem's hypothesis and is settled by the full claims.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Canadian Journal of Mathematics, volume 18 (1966), pp. 643--655, by the publisher's record. The site's curator, T. F. Bloom, mentions the theorem in the problem's commentary as the earlier result under the stronger assumption, but the site's PROVED label credits Szemerédi and Vu, so the commentary is not listed as reviewed evidence. This claim is partial, so the problem's standing derives from the full claims. Nothing here was checked by this project.