Dense clique family blow-up lemma ($|\mathcal{M}|\ge \eta n^s⟹K_s(\lfloor a\log n\rfloor )\subseteq G$)

A family of positively many $s$-cliques forces a logarithmic balanced graph blow-up, which can also contain a matching of that many members of the family. To induct on $s$, prune faces with few extensions, apply induction to the remaining $(s-1)$-faces, and use a matching of those faces as one side of a bipartite incidence graph. The common neighbourhood from bipartite density estimate selects logarithmically many disjoint faces with polynomially many common extension vertices.