For $c>0$ let $f_c(n)$ be the largest size of a set $A\subseteq\{1,\dots,\lfloor cn\rfloor\}$ such that $n$ is not a sum of distinct elements of $A$. Erdős and Graham asked whether $f_c(n)$ depends on $n$ in an irregular way (Erdős Problem #361). We prove that $f_c(n)=\lfloor cn\rfloor-\lceil n/2\rceil$ for $c\ge 1$, while for $0<c<1$ the ratio $f_c(n)/n$ has $\limsup$ equal to $c/2$, attained along odd $n$, and, along the integers $n$ whose least non-divisor equals a prime power $q$, tends to \[ \max\Bigl\{\frac cq,\; c-\frac12+\frac{1-c}{q}\Bigr\} \qquad\text{whenever } 2^{q-2}\min\{c,1-c\}\le 1 . \] Consequently $f_c(n)/n$ converges if and only if $c\ge1$; for $0<c<1$ it has at least two limit points, and arbitrarily many as $c\to0^+$ or $c\to1^-$. For $c=1/2$ the set of limit points is exactly $\{1/6,1/4\}$. The upper bounds follow from an inverse theorem of independent interest: if $s\ge1$, $q$ is the least positive integer not dividing $s$, and $\varepsilon>0$, then every $A\subseteq\{1,\dots,E\}$ with $|A|\ge E/q+\varepsilon T$ has a subset summing to $sT$, uniformly for all sufficiently large $T$ and all $E\le sT/2^{q-2}$. For $s=2$, this extends Alon's theorem on subsets of $\{1,\dots,N\}$ with no subset sum equal to $2N$. The results for $c\ge1$ and the $\limsup$, as well as the limits and the inverse theorem for $q\in\{2,3\}$, have been formalized and machine-checked in the Lean 4 proof assistant.