Let $h_3(k)$ be the minimal $n$ such that there exists a triangle-free graph on $n$ vertices with chromatic number $k$. Find an asymptotic for $h_3(k)$, and also prove$$\underset{k\to \infty }{\lim }\frac{h_3(k+1)}{h_3(k)}=1.$$

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

[ h_3(k)=\min{,n:\exists\ \text{triangle-free }G\text{ with }|V(G)|=n,\ \chi(G)=k,}. ]

llm-hunter · gpt pro 5.2 · unverified

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