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Let a fixed simple three-uniform hypergraph $F$ have $m$ hypergraph edges, with one host vertex of a hypergraph class $V_i$ for each vertex of a hypergraph. Assume each hypergraph edge triple has indicator $h_e$, subset density $p_e$, and balanced three-dimensional box norm at most ${\alpha }^{1/8}$. The normalized labelled hypergraph copy count is $\mathbb{E}{\prod }_{e\in E(F)}{h}_e$. Telescope this product against ${\prod }_e{p}_e$. In each error term, fix all variables outside the three vertices of a hypergraph of its balanced hypergraph edge. Every other hypergraph edge intersects that triple in at most two vertices of a hypergraph, so the remaining factors group into three pair functions bounded by one. The pair-factor correlation bound for the three-dimensional box norm bounds the error by ${\alpha }^{1/8}$. Summing givesFor the three-uniform tetrahedron, $m=4$. The same argument with a common host set counts homomorphisms; excluding collisions of distinct abstract vertices of a hypergraph changes the count by at most $\left(\genfrac{}{}{0ex}{}{|V(F)|}{2}\right)|V{|}^{|V(F)|-1}$.

First prove the pair-factor correlation bound for the three-dimensional box norm. For real $g$ and three functions $a(y,z),b(x,z),c(x,y)$ bounded in absolute value by one, letApply the Cauchy-Schwarz inequality in $(y,z)$, placing the $x$ average inside and removing $a$. This givesExpand this square, with independent $x_0,x_1$. Apply Cauchy-Schwarz in $(x_0,x_1,z)$, removing the product $b(x_0,z)b(x_1,z)$. The result isExpand with independent $y_0,y_1$ and apply Cauchy-Schwarz in $(x_0,x_1,y_0,y_1)$, removing the four $c$ factors. ThenTherefore $|T|\le \Vert g{\Vert }_{{\square }^3}$, with no independence assumption on the three pair functions.

In this multipartite hypergraph, a simplex is a three-uniform tetrahedron: one vertex of a hypergraph from each of $X,Y,Z,W$, with all four possible triples present. Write the corresponding hypergraph edge indicator functions as $h_1(x,y,z),h_2(x,y,w),h_3(x,z,w),h_4(y,z,w)$, of subset densities $p,q,r,s$. Its normalized count isTelescope the product exactly:For the first term fix $w$. The remaining three factors are bounded pair functions of $(x,y)$, $(x,z)$ and $(y,z)$, so the inequality just proved bounds its $(x,y,z)$ average by $\Vert h_1-p{\Vert }_{{\square }^3}\le {\alpha }^{1/8}$. Averaging over $w$ preserves that bound. For each other term fix the vertex of a hypergraph outside its balanced triple and apply the same argument, allowing an unused pair factor to be identically one. The subset density prefactors are at most one. HenceMultiplying by $|X||Y||Z||W|$ givesThus one may take the absolute constant $C=4$. This proves the complete-pair-support case of the tetrahedron counting lemma directly; no lower bound on $p,q,r,s$ is needed.