 Logarithmic Erdős-Stone theorem

An edge density exceeding the Turan theorem threshold for $K_{r+1}$ by a fixed positive amount forces a balanced $K_{r+1}(t)$ subgraph with logarithmic part size. Combine clique supersaturation by sampling with the dense clique family blow-up lemma. Dense binomial random graphs show that the logarithmic order is optimal for a uniform density guarantee.