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 $T$?