Let $A\subset \mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c\in A$ (aside from the trivial coincidences). Is it true that$$\mathrm{liminf}\frac{|A\cap \{1,\dots ,N\}|}{N^{1/3}}=0?$$

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

Your condition is exactly what people call a **(B_3)** sequence (a “3‑Sidon set”): all sums $a+b+c$ (with order ignored) are different. Erdős asked this as a specific open problem (it is listed as **open** on the Erdős Problems site). ([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.