Logarithmic clique blow-up from minimum degree ($\delta (G)\ge (1-1/r+\epsilon )n⟹K_{r+1}(\lfloor d\log n\rfloor )\subseteq G$)

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.