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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 nondecreasing sequence $A=\{a_1\le a_2\le\cdots\}$ of positive integers with $a_n\le Mn^{\alpha}$ for all $n$, for some $M$ and some $0\le\alpha<1$ (the paper's condition (1.1)), is subcomplete, that is, its finite subset sums contain an infinite arithmetic progression. The paper's Theorems 1.1 and 1.2 deduce completeness from it under a residue condition; the source card records the statements. The hypothesis is the case of Problem 343 in which the counting function $\lvert A\cap\{1,\ldots,N\}\rvert$ is $\gg N^{1+\epsilon}$ for some $\epsilon>0$, which the site's remarks credit to the paper.

Covers. The problem's multisets whose counting function is $\gg N^{1+\epsilon}$ for some $\epsilon>0$, equivalently $a_n\le Mn^{\alpha}$ with $\alpha<1$; such a counting function exceeds $CN$ for all large $N$, so these multisets are a special case of the hypothesis of the corrected Statement, and for them the answer is yes. It does not cover a multiset with at least $CN$ terms up to $N$ for all large $N$ whose counting function is not $\gg N^{1+\epsilon}$ for any $\epsilon>0$. The paper's companion construction, a multiset with counting function $\gg N^{1-\epsilon}$ that is not subcomplete, settles no instance of the question either, since its counting function is not linear.

Acceptance. Refereed: the paper is a journal article in Canad. J. Math. The site's remarks credit the result, but the site's label credits Szemerédi and Vu, so no reviewed evidence is listed. Crossref gives the year only, so the page is dated to the first day of 1966. The proof is not reviewed here.

Depends on. No page of this wiki.