Labelled blowup and good induced copy

Let $J$ be a nonempty finite simple graph with its vertices regarded as labels. For an integer $t\ge 1$ and $0\le q\le 1$, a $(t,q)$-blowup of $J$ in $G$ is a family of pairwise disjoint sets $(A_j:j\in V(J))$, each of size $t$, with the following property: for distinct $i,j$, each vertex of $A_i$ has at most $qt$ neighbors in $A_j$ if $ij\mathrm{\notin }E(J)$, and at most $qt$ nonneighbors in $A_j$ if $ij\in E(J)$. The condition is required for both ordered pairs $(i,j)$ and $(j,i)$.

For $I\subseteq V(J)$, a good embedding of $J[I]$ is an induced embedding $\varphi$ satisfying $\varphi (i)\in A_i$ for every $i\in I$. Counts mean labelled embeddings as in Induced copy density and homogeneous restriction parameter. The empty map is good when $I=\mathrm{\varnothing }$. Internal edges of a block are unrestricted.

The family consists of blocks in the sense of Blockades, their length, their width, and their support, with both directional conditions of Sparsity of one vertex set to another, and weak sparsity of a pair. Merely meeting distinct blocks does not impose the specified label assignment.

Proof/convention locator: Bucic, Nguyen, Scott and Seymour, Induced subgraph density I, Section 4, definition preceding 4.2.