Erdős Problem #1009

Is it true that, for every \(c>0\), there exists \(f(c)\) such that every graph on \(n\) vertices with at least \(\lfloor n^2/4\rfloor+k\) edges, where \(k<c n\), contains at least \(k-f(c)\) many edge disjoint triangles? Erdős [Er71] proved this for \(c<1/2\) using a theorem of Erdős and Gallai, which says that every graph on \(n\) vertices with at least \((n-1)^2/4+2\) many edges, with chromatic number \(3\), must contain a triangle. In fact, Erdős proved this is true with \(f(c)=0\) for \(c<1/2\). At first Erdős thought \(f(c)=0\) for larger values of \(c\) but this is false: an example of Sauer proves that \(f(2)\geq 1\). This is true, and was proved by Győri [Gy88] who proved that this is true with \(f(c)\ll c^2\), and also that \(f(c)=0\) if \(c<2\) for odd \(n\) or \(c<3/2\) for even \(n\). A family of edge disjoint triangles is a finite set of \(3\)-cliques of \(G\) any two of which share at most one vertex.