A triad is the set of triples supported on one prescribed pair cell for each of the three pairs of vertex classes. Relative triple density is measured among those supported triples. In strong hypergraph regularity, the pair cells must themselves be sufficiently regular and have controlled density before a small relative three-dimensional box norm can give a tetrahedron counting lemma.
A sufficiently regular four-partite three-uniform hypergraph, supported on sufficiently regular pair cells of positive density and with positive relative triple densities, has a positive fourth-power number of three-uniform tetrahedra. On complete pair supports, telescoping the four edge functions shows that an error of at most in each three-dimensional box norm gives a counting error at most . The relative version additionally requires control of the pair-support densities.
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