Strong regularity for three-uniform hypergraphs

Strong regularity refines both the vertex classes and the pair graphs supporting triple hyperedges. The resulting triads in a hypergraph regularity partition have nearly constant relative triple density outside a small exceptional weight. Pair and triple errors must be chosen in a hierarchy strong enough for a relative tetrahedron counting lemma; ordinary graph regularity on the vertex partition alone is insufficient. Refinement raises bounded conditional-density energies.

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 $\xi$ in each three-dimensional box norm gives a counting error at most $4\xi$. The relative version additionally requires control of the pair-support densities.