Erdős problem 1034

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 $T$?

A conjecture of Erdős and Faudree; a stronger version of [905].

This has been solved in the negative by Ma and Tang [MaTa25], who construct a graph with $n$ vertices and $>n^2/4$ edges in which every triangle has at most $(2-(5/2)^{1/2}+o(1))n$ vertices adjacent to at least two of its vertices (note that $2-(5/2)^{1/2}\approx 0.4189$).