Erdős Problem #42 — Sidon sets difference-disjoint from a maximum Sidon set

The problem asks whether every Sidon set $A$ in $\{1,\ldots,N\}$ eventually admits an $M$-element Sidon set $B$ with no nonzero difference in common. The cases $M=1$ and $M=2$ are settled, and a full proof has been claimed but remains under examination.

Known results

• $M=1$ and $M=2$: settled; the source gives no mathematician or year.
• $M=3$: proved by Sedov using ChatGPT and Codex.
• Sparse regime: for fixed $M$ and Sidon $A$ with $|A|=o(\sqrt N)$, the assertion holds for sufficiently large $N$.
• A claimed quantitative extension gives $|B|\gg\left(\frac{\log\log N}{\log\log\log N}\right)^{1/2}$; its proof is not verified.

Claimed full solution

Sandhu prompted GPT 5.5 Pro to produce a proof for all $M$. A separate Fourier--compactness write-up states the same theorem, but provides neither independent verification nor publication details; the Lean formulation still displays sorry.

Current status (as of February 2026): $M=1$, $M=2$, $M=3$, and a sparse-set subcase are recorded, while the claimed all-$M$ solution remains unverified.