Problem 301

Erdős Problem #301: declared status 'open'. Formalized: no. Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to$$\frac{1}{a}\neq \frac{1}{b_1}+\cdots+\frac{1}{b_k}$$with distinct $a,b_1,\ldots,b_k\in A$? Estimate $f(N)$. In particular, is $f(N)=(\tfrac{1}{2}+o(1))N$? Current best: All such $S_a$ are disjoint and, if $A$ has no solutions to the given equation, then $A$ must omit at least two elements of $S_a$ when $a\leq N/12$ and at least one element of $S_a$ when $N/12<a\leq N/6$, and an elementary calculation concludes the proof. Prize: no. OEIS: A390394. Tags: number theory, unit fractions.