Let $N\ge 1$ and $A\subset \{1,\dots ,N\}$ be a Sidon set. Is it true that, for any $ϵ>0$, there exist $M$ and $B\subset \{N+1,\dots ,M\}$ (which may depend on $N,A,ϵ$) such that $A\cup B\subset \{1,\dots ,M\}$ is a Sidon set of size at least $(1-ϵ)M^{1/2}$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

This is **not known** in general. It is a well known Erdős question (listed as **Erdős Problem #44: “extending Sidon sets”**) and it is still marked open. ([Erdős Problems][1])

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

3 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.