Packing cliques in graphs with independence number 2

Let G be a graph with no three independent vertices. How many edges of G can be packed with edge-disjoint copies of K_k More specifically, let f_k(n, m) be the largest integer t such that, for any graph with n vertices, m edges, and independence number 2, at least t edges can be packed with edge-disjoint copies of K_k. Turán's theorem together with Wilson's Theorem assert that f_k(n, m) = (1 -o(1))n²/4 if m ≈ n²/4. A conjecture of Erdos states that f₃(n, m) > (1 - o(1))n²/4 for all plausible m. For any e > 0, this conjecture was open even if m ≤ n²(1/4 + e). Generally, f _k(n, m) may be significantly smaller than n²/4. Indeed, for k = 7 it is easy to show that f₇(n, m) ≤ 21/90 n² for m ≈ 0.3n². Nevertheless, we prove the following result. For every k ≥ 3 there exists γ > 0 such that if m ≤ n²(1/4 + γ) then f_k(n, m) ≥ (1 - o(1)))n²/4. In the special case k = 3 we obtain the reasonable bound γ ≥ 10^-4. In particular, the above conjecture of Erdos holds whenever G has fewer than 0.250.1n² edges.