If \(G\) has no \(5\)-cliques, and any large subset of vertices contains a triangle, then \(G\) is sparse

Problem [1]

Suppose a graph \(G\) contains no \(K_5\) and for every \(\epsilon >0\) and \(n \geq n_0(\epsilon)\), every set of \(\epsilon n\) vertices contains a triangle. Is it true that \(G\) can only have \(o(n^2)\) edges?

The best available result [1] is that such \( G\) contains at most \( \frac 1 {12} n^2 (1+o(1))\) edges.

1 P. Erdös, Problems and results in combinatorial analysis and combinatorial number theory, Graph theory, combinatorics and applications, Vol. 1 (Kalamazoo, MI, 1988), Wiley-Intersci. Publ., 397-406, Wiley, New York, 1991.