Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$ and vertices $y_1,\ldots,y_t$, where $t>(\frac{1}{2}-o(1))n$, such that every $y_i$ is joined to at least two vertices of $T$?

Erdős and Faudree asked about the threshold $h(n)$ such that every graph with $n$ vertices and $>n^2/4$ edges contained a triangle and $h(n)$ other vertices which are connected to at least two vertices of the triangle. The fact that every graph with $>n^2/4$ edges contains a book of size $n/6$ shows that $$(1/6-o(1))n \leq h(n).$$