= Labelled hypergraph copy count
For <hypergraphs> $F,H$, a labelled copy is an <injective function> $\phi:V(F)\to V(H)$ taking every <hypergraph edge> of $F$ to an <hypergraph edge> of $H$. Specified <vertex of a hypergraph> classes may additionally require $\phi(i)\in V_i$. The count is the sum, over these <injective> maps, of the product of the host <hypergraph edge> indicators. Without injectivity the same expression counts <hypergraph> homomorphisms. On a common host of size $n$, their counts differ by at most $\binom{|V(F)|}{2}n^{|V(F)|-1}$: every noninjective map identifies at least one pair of abstract <vertices of a hypergraph>, each prescribed equality permits at most $n^{|V(F)|-1}$ maps, and the <union bound> completes the estimate. An induced copy additionally preserves nonedges, so its indicator product includes nonedge factors. The <counting lemma for octahedrally quasirandom three-uniform hypergraphs> estimates these labelled products when the requisite centered triple indicators have small box <norm>.
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