= Octahedral quasirandomness of a three-uniform hypergraph
{title2=$\|1_H-p\|_{\square^3}^8\leq\alpha$}
For a tripartite three-<uniform hypergraph> on nonempty finite parts $X,Y,Z$, let $h:X\times Y\times Z\to\{0,1\}$ be its <hypergraph edge> indicator, let $p=\mathbb Eh$ and put $g=h-p$. In the eighth-power convention, $\alpha$-quasirandomness means
$$
\|g\|_{\square^3}^8=\mathbb E_{x_0,x_1,y_0,y_1,z_0,z_1}\prod_{i,j,k\in\{0,1\}}g(x_i,y_j,z_k)\leq\alpha.
$$
All averages are uniform and independent, with repeated sampled <vertices of a hypergraph> allowed. This expression is nonnegative because it equals $\mathbb E_{x_0,x_1,y_0,y_1}(\mathbb E_z\prod_{i,j}g(x_i,y_j,z))^2$. It measures correlations across the eight faces of a tripartite octahedral configuration. If instead the <norm> itself is used as the quasirandomness parameter, the parameter in this definition is its eighth power.
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