Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\dots ,N\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that$$H(k{)}^{1/k}/k\to \infty$$as $k\to \infty$?

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

Write $W(k,r)$ for the usual van der Waerden number: the least $N$ such that every $r$-colouring of $[N]=\\{1,\dots,N\\}$ contains a monochromatic $k$-term arithmetic progression.

llm-hunter · gpt pro 5.2 · unverified

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