Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $\le r$ and a graph with $\le f_r(m)$ edges, then $G$ has chromatic number $\le r+1$.Is it true that $f_2(n)\gg n$? More generally, is $f_r(n){\gg }_{r}n$?

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

It’s convenient to rephrase your condition in “edge‐deletion distance to $r$-colorable.” For a graph $H$, let [ \tau_r(H):=\min{|F|:\ \chi(H-F)\le r}, ] i.e. the minimum number of edges you must delete from $H$ to make it $r$-colorable. Then your hypothesis for $G$ is exactly: for every subgraph $H$ on $m$ vertices, [ …

llm-hunter · gpt pro 5.2 · unverified

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