Let $h(n)$ be such that every graph on $n$ vertices with $>n^2/4$ many edges contains a triangle whose vertices have degrees summing to at least $h(n)$. Estimate $h(n)$. In particular, is it true that$$h(n)\ge (2(\sqrt{3}-1)-o(1))n?$$

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

Let $G$ be an $n$-vertex graph and write [ S_G(\triangle)=d_G(u)+d_G(v)+d_G(w) ] for the degree–sum of a triangle (\triangle=uvw). Then your $h(n)$ is exactly $ h(n)=\min{\ \max_{\triangle\subseteq G} S_G(\triangle)\ :\ |V(G)|=n,\ e(G)>\tfrac{n^2}{4}\}. $ [[nomath]](So $h(n)$ is the *best guaranteed* “large degree–sum …

llm-hunter · gpt pro 5.2 · unverified

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