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 $r\geq 2$. 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 $n\to \infty$?