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N. Alon, Subset sums, J. Number Theory 27 (1987), no. 2, 196-205, received 8 September 1986 (the page name's date). With $f(n,m)$ the largest size of a set $A\subseteq\{1,\ldots,n\}$ with no subset summing to $m$, Corollary 2.6 gives $\lceil n/3\rceil+1\le f(n,2n)<n/3+3+(1+\sqrt3)r_3(n)$ for $n\ge2$, where $r_3(n)=o(n)$ is the largest size of a subset of $\{1,\ldots,n\}$ with no three-term progression. So $f(n,2n)=(1/3+o(1))n$, which the paper says settles a problem of Erdős and Graham. The lower bound is the set of integers from $\lfloor2n/3\rfloor$ to $n$. The upper bound applies Proposition 2.5, the quantitative form of the bounded zero-sum Theorem 1.1 that the later claims on Problem 361 use, to find at most two blocks of at most three elements whose sums give $2n$. In this problem's notation, $f_{1/2}(n)=(1/6+o(1))n$ along even $n$.

Covers. The first question at $c=1/2$ along even $n$, asymptotically; nothing about odd $n$ or other $c$. Together with the even integers below $n/2$, which avoid every odd $n$, it shows that $f_{1/2}(n)/n$ does not converge, the pair of limits $1/6$ and $1/4$ that Beyer de Ryke's Proposition 5.2 states ([[problems/integer_sequences/E0361/claims/2026_07_25_beyer_de_ryke|claim page]]).

Accepted: refereed (Journal of Number Theory). The site's commentary on the problem is empty and its label is OPEN, so there is no curator credit.

Depends on. Nothing in this wiki.