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For hypergraphs $F,H$, a labelled copy is an injective function $\varphi :V(F)\to V(H)$ taking every hypergraph edge of $F$ to an hypergraph edge of $H$. Specified vertex of a hypergraph classes may additionally require $\varphi (i)\in V_i$. 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 $\left(\genfrac{}{}{0ex}{}{|V(F)|}{2}\right)n^{|V(F)|-1}$: 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.

Yes. The same telescoping argument proves the counting lemma for octahedrally quasirandom three-uniform hypergraphs for every fixed simple three-uniform hypergraph $F$.

Label its vertices of a hypergraph $1,\dots ,v$ and give vertex of a hypergraph $i$ a nonempty host part $V_i$. For each hypergraph edge $e=\{i,j,k\}$, let $h_e$ be the corresponding host hypergraph edge indicator function of subset density $p_e$, with $\Vert h_e-p_e{\Vert }_{{\square }^3}\le {\alpha }^{1/8}$. The normalized labelled hypergraph copy count isIf the host parts are disjoint, every such vertex of a hypergraph choice is automatically injective. Enumerate the $m$ hypergraph edges as $e_1,\dots ,e_m$ and telescopeIn the term for $e_j=\{a,b,c\}$, fix all variables outside this triple. Since $F$ is simple and three-uniform, every other hypergraph edge intersects $\{a,b,c\}$ in at most two vertices of a hypergraph. Its remaining factor therefore depends on at most two of $x_a,x_b,x_c$. Group all two-variable factors according to the pairs $(a,b),(a,c),(b,c)$. Absorb each one-variable factor into any pair containing its variable, and constants into any group. The three resulting pair functions remain bounded by one.

The balanced factor is now multiplied by exactly the sort of pair functions handled in part (ii). Its average has absolute value at most ${\alpha }^{1/8}$, independently of the fixed outside variables. Averaging those variables and summing the $m$ errors givesEquivalently,This counts labelled copies preserving the specified vertex of a hypergraph classes, for arbitrary intersection patterns among the hypergraph edges, and includes the three-uniform tetrahedron as a special case.

For a common host vertex of a hypergraph set of size $n$, the same product average counts homomorphisms. At most $\left(\genfrac{}{}{0ex}{}{v}{2}\right)n^{v-1}$ assignments identify a pair of distinct labelled vertices of a hypergraph, by the union bound. Thus restricting to injective copies changes the estimate by at most this amount. With a common normalized subset density $p$ the injective count is therefore $p^m{n}^v+O(m{\alpha }^{1/8}{n}^v+v^2{n}^{v-1})$. For induced copies, include the nonhypergraph edge indicator functions as additional factors; their balanced functions are negatives of the corresponding edge-balanced functions and have the same box norm. The same proof applies provided all the relevant triple parts satisfy the stated quasirandomness condition.