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Claim. Let $c(N)$ be the largest size of a set $A\subseteq\{1,\ldots,N\}$ carrying signs $\delta:A\to\{-1,1\}$ whose signed reciprocals sum to zero while no nonempty proper subset of $A$ sums to zero, as Problem 319 defines it. The site's commentary credits Sarosh Adenwalla with the observation that

$$c(N)\ge\Bigl(1-\frac1e+o(1)\Bigr)N$$

follows from the Main Theorem of Croot's paper On unit fractions with denominators in short intervals, Acta Arith. 99 (2001), no. 2, 99--114, whose card is croot_1999_unit_fractions_denominators_short_intervals. In outline: Croot's theorem gives a set $B$ of integers in $[(\frac1e-o(1))N,N]$ with $\sum_{b\in B}1/b=1$; all the integers of that interval have reciprocal sum $1+o(1)$ and each one omitted removes at least $1/N$, so $|B|\ge(1-\frac1e-o(1))N$; and the signs $\delta(1)=1$ and $\delta(b)=-1$ for $b\in B$ make $A=B\cup\{1\}$ a signed zero-sum set. The problem page writes out the minimality check: a proper nonempty subset of $A$ either omits $1$, so its signed sum is negative, or contains $1$ and misses some element of $B$, so its signed sum is positive.

Covers. The lower bound $c(N)\ge(1-\frac1e+o(1))N$ only. With the trivial upper bound $c(N)\le N$ it gives $c(N)$ the order $N$, which answers the $\Theta$-order variant posed in the formal-conjectures statement file, but it does not determine the asymptotic of $c(N)$: the limit of $c(N)/N$, if it exists, is left anywhere in $[1-\frac1e,1]$. The claim's value is proved because the result proves a bound.

Standing. Claimed. The result exists only as the site's commentary: no written source by Adenwalla states it, and it has no arXiv version, no journal record, no formalization and no independent review. The commentary credits it to Adenwalla, but the site labels the problem OPEN (no last-edited stamp; OPEN on 2026-10-07) and lists no parts, so the credit is not an acceptance and no reviewed evidence is listed. Croot's theorem itself is refereed, but the deduction is not published. The claim is dated by the earliest archived copy of the site's page that carries the remark, that of 15 September 2025 (the second link); the archived copy of 19 June 2024 carries the problem without commentary. The pending density-one claim of 16 July 2026 would lift this bound to $c(N)\ge N-o(N)$.

Depends on. Croot's Main Theorem.