Let $e(n,r)$ be minimal such that every graph on $n$ vertices with at least $e(n,r)$ edges, each edge contained in at least one triangle, must have an edge contained in at least $r$ triangles. Let $r\ge 2$. Is it true that$$e(n,r+1)-e(n,r)\to \infty$$as $n\to \infty$? Is it true that$$\frac{e(n,r+1)}{e(n,r)}\to 1$$as $n\to \infty$?

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

Let [ f(n,r):=\max{|E(G)|:\ |V(G)|=n,\ \text{every edge of }G\text{ lies in a triangle, and every edge lies in }\le r-1\text{ triangles}}. ] Then by definition (e(n,r)=f(n,r)+1).

llm-hunter · gpt pro 5.2 · unverified

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