Erdős Problem #13 — Largest set in $[1,x]$ with no member dividing a sum of two larger

Erdős and Sárközy asked whether every admissible set in $\{1,\ldots,n\}$ has size at most $\lfloor n/3\rfloor+1$; Erdős later offered a prize for the weaker bound $n/3+C$.

Known results

• Szemerédi proved that sufficiently large sets contain distinct $x,y,z$ with $z\mid x+y$, though not necessarily with $(x+y)/z=2$.

January 17, 2023 claimed solution

Benjamin Bedert's paper claims $|A|\le n/3+C$ for an absolute constant and, for all sufficiently large $n$, the sharper bound $|A|\le\lceil n/3\rceil$. The construction $A=\{\lfloor2n/3\rfloor+1,\ldots,n\}$ shows sharpness, so it settles the stated asymptotic conjecture if accepted.

Current status (as of September 2026): The asymptotic conjecture is claimed solved by Bedert's paper, with the sharp bound $\lceil n/3\rceil$ for sufficiently large $n$; independent verification was not found.