Logarithmic Erdős-Stone theorem ($t\ge \lfloor d(r,ϵ)\log n\rfloor$)

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.

A fixed positive excess over the Turan theorem threshold forces logarithmic balanced complete multipartite graphs. There is an elementary induction using common neighbourhood from bipartite density: first obtain a logarithmic $r$-partite subgraph from the induction hypothesis, and select disjoint transversal $r$-cliques in it. Each such clique has at least $r\epsilon n$ common neighbours, by the union bound on non-neighbour sets. Apply the bipartite common-neighbour count to these cliques as left-side objects. Their selected union and a large common neighbourhood form the extra part.

In a binomial random graph with fixed edge probability below one, the displayed first-moment bound tends to zero for $t=C\log n$ and sufficiently large fixed $C$. At the same time the edge density concentrates near $p$. Thus a positive fixed density cannot force balanced complete multipartite subgraphs whose part size grows faster than logarithmically.