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\"{o}dl, 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: TH for the size of the vertex component, and LH for the size of the pairs component. In this paper we consider the behavior of the function LH. We show any such function is either constant, bounded above and below by a polynomial, or bounded below by an exponential.