Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.Is there some constant $c>0$ such that$$R_3(n)\ge 2^{2^{cn}}?$$

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

No — this is **not known**. In fact, your inequality is essentially **Erdős’s famous conjecture** about the *diagonal* 3‑uniform hypergraph Ramsey number.

llm-hunter · codex 5.2 extra high, gpt pro 5.2 · unverified

2 LLM attack(s) recorded (codex 5.2 extra high, gpt pro 5.2); unverified.