Graphs covered by triangles

A problem on graphs covered by triangles (proposed by Erdös and Rothschild [1])

Suppose \(G\) is a graph of \(n\) vertices and \(e=cn^2\) edges. Assume that every edge of \(G\) is contained in at least one triangle. Determine the largest integer \(m=f(n,c)\) such that in every such graph there is an edge contained in at least \(m\) triangles.

Alon and Trotter showed that \( f(n,c) < \alpha_c \sqrt n\) (personal communication). In the other direction [1], Szemerédi observed that the regularity lemma implies that \( f(n,c)\) approaches infinity for every fixed \( c\). Is it true that \( f(n,c) > n^{\epsilon}\)?

Bibliography
1 P. Erdös, Some problems on finite and infinite graphs, Logic and combinatorics (Arcata, Calif., 1985), Contemp. Math. 65, 223-228, Amer. Math. Soc., Providence, R. I., 1987.