Erdős Problem #888 — What is the size of the largest $A\subseteq \{1,\ldots,n\}$ such that if $a\leq b\leq c\leq d\in A$ are such that $abcd$ is a square then $ad=bc$?

The problem asks for the largest admissible subset of $[1,n]$, conjectured to have order $n\log\log n/\log n$. The formal Lean statement remains incomplete, with the main theorem containing sorry.

Known results

• Sárközy proved an $o(n)$ upper bound; Tao later supplied a proof.
• The primes give a lower bound of order $n/\log n$.
• The primes together with squarefree semiprimes give $(1+o(1))n\log\log n/\log n$.

May 2026 claimed upper bound

The site attributes to GPT-5.5 Pro, prompted by Chojecki, a colored bipartite-graph argument claiming the matching upper bound $|A|\ll n\log\log n/\log n$. If correct, this settles the conjectured order, but the proof remains unverified and no independent mathematical publication or check is recorded.

Current status (as of September 2026): The lower construction and earlier $o(n)$ upper bound are established, while the matching upper bound is claimed but remains unverified; the problem is not formally resolved.