Tuza's Conjecture under an Edge-Count Condition

For a graph $G$, let $\nu (G)$ denote the maximum number of pairwise edge-disjoint triangles and let $\tau (G)$ denote the minimum number of edges meeting every triangle. Tuza’s conjecture asserts that $\tau (G)\le 2\nu (G)$ for every graph $G$. We first note that Tuza’s conjecture is equivalent to the following statement: for every fixed $c\ge 0$, if $G$ is a sufficiently large $n$-vertex graph with $e(G)\ge n^2\mathrm{/}4-cn$, then $\tau (G)\le 2\nu (G)$. In this paper, we prove this statement for every $c<2$.