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Claim. For every non-constant sequence $\delta_1,\delta_2,\ldots$ of $\pm1$'s there is a finite non-empty set $S$ of positive integers with $\sum_{i\in S}\delta_i/(2i+1)=0$: the odd numbers from $3$, which the site calls the odd numbers, have property $P_1$. This is one progression of the first question of Problem 318. The paper is not held in the library, and the statement follows printed p. 42 of the 1980 monograph of Erdős and Graham, which calls it the more difficult counterpart of Erdős and Straus's case of the positive integers.

Covers. The one progression $\{2i+1:i\ge1\}$. It does not settle the arithmetic-progression question in general, the positive-density question or the squares question.

Depends on. No page of this wiki.

Source and acceptance. R. Sattler, Solution to Problem 387, Nieuw Arch. Wisk. (3) 23 (1975), 184--189: a solution published in a journal, which is the refereed evidence; the volume has no DOI and no online copy is known, so the paper is cited here and only the site's page is linked. The site's curator credits the result in the problem's commentary, but the result settles only one progression and not the part, so that credit is not listed as reviewed. The result is contained in Sattler's 1982 theorem that every infinite arithmetic progression has property $P_1$, Sattler's theorem on arithmetic progressions.

Dating. The page is dated by the publication year; the volume gives no day, and the day in the page name is a placeholder.