Let $R(n;k,r)$ be the smallest $N$ such that if the edges of $K_N$ are $r$-coloured then there is a set of $n$ vertices which does not contain a copy of $K_k$ in at least one of the $r$ colours. Prove that there is a constant $C=C(r)>1$ such that$$R(n;3,r)<C^{\sqrt{n}}.$$

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

The inequality you ask for is **not true** (with your definition of (R(n;k,r))). In fact, already for (r=2) one can build 2‑colourings where **every** (n)-set contains **both** a red triangle and a blue triangle, as long as (N) is of the form (C^n). So (R(n;3,2)) grows at least exponentially in (n), and therefore it ca…

llm-hunter · gpt pro 5.2 · unverified

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