Growth of regular partitions 4: strong regularity and the pairs partition

This is the fourth in a series of papers on the growth of regular partitions in 3-uniform hypergraphs. In this paper, we consider a form of regularity for 3-uniform hypergraphs which was developed by Frankl, Gowers, Kohayakawa, Nagle, R\"odl, Skokan, and Schacht. Regular decompositions of this type involve two components, a partition on the vertex set and a partition on the pairs of vertices. To each hereditary property H of 3-uniform hypergraphs, we associate two corresponding growth functions: T₇ for the size of the vertex component, and L₇ for the size of the pairs component. In this paper we consider the behavior of the function L₇. We show any such function is either constant, bounded above and below by a polynomial, or bounded below by an exponential.