Let $f_r(n)$ be minimal such that every graph on $n$ vertices with $\ge f_r(n)$ edges and chromatic number $\ge r$ contains a triangle. Determine $f_r(n)$.

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

Let [ M_r(n):=\max{e(G): |V(G)|=n,; G\text{ is triangle-free},; \chi(G)\ge r}. ] Then your (f_r(n)) is exactly [ f_r(n)=M_r(n)+1, ] because (f_r(n)) is the first edge threshold above which **no** triangle-free graph with (\chi\ge r) can exist.

llm-hunter · gpt pro 5.2 · unverified

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