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 Kk? More specifically, let fk(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 ofKk. Turán’s Theorem together with Wilson’s Theorem assert that fk(n,m) = (1−o(1)) 2 4 ifm ≈ n2 4 . A conjecture of Erdős states that f3(n,m) ≥ (1−o(1)) n2 4 for all plausible m. For any > 0, this conjecture was still open even if m ≤ n( 1 4 + ). Generally, fk(n,m) may be significantly smaller than n 2 4 . Already for k = 7 it is easy to show that f7(n,m) ≤ 21 90n 2 for m ≈ 0.3n. Nevertheless, we prove the following result. For every k ≥ 3 there exists γ > 0 so that if m ≤ n( 1 4 + γ) then fk(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 Erdős holds whenever G has less than 0.2501n edges.