Graph blow-up ($H(t)$)

A blow-up replaces each vertex of a graph by an independent set, and each original edge by all edges between the corresponding two sets. Replacing every vertex of $K_r$ by $t$ vertices gives the balanced complete multipartite blow-up $K_r(t)$.

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.

Cloning a vertex replaces it by several nonadjacent vertices with identical neighbourhoods. In homomorphism density, conditioning on the images of all other vertices reduces cloning to the inequality $\mathbb{E}{p}^t\ge (\mathbb{E}p{)}^t$ from the Jensen inequality. Iterated cloning shows that positive clique homomorphism density forces any fixed complete multipartite blow-up.