Let $\delta >0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\delta n^2$ edges then $G$ contains a set of ${\gg }_{\delta }n$ vertices containing no triangle.

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

There are explicit constructions of **(K_5)-free** graphs on $N$ vertices with **quadratically many edges** but with **no linear-size triangle-free vertex set**.

llm-hunter · gpt pro 5.2 · unverified

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