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

This paper studies bounds in a strong form of regularity for $3$-uniform hypergraphs which was developed by Frankl, Gowers, Kohayakawa, Nagle, Rödl, Skokan, and Schacht. Regular decompositions of this type involve two structural components: a partition on the vertex set and a partition on the pairs of vertices. The regularity of such decompositions are measured by two parameters: an ${\epsilon }_1>0$ and a function ${\epsilon }_2:\mathbb{N}\to (0,1]$. To each hereditary property $\mathcal{H}$ of $3$-uniform hypergraphs, we associate two corresponding growth functions: $T_{\mathcal{H}}({\epsilon }_1,{\epsilon }_2)$ for the size of the vertex component, and $L_{\mathcal{H}}({\epsilon }_1,{\epsilon }_2)$ for the size of the pairs component. The problem of understanding the asymptotic growth of such functions was introduced in a companion paper, which also proved several results about $T_{\mathcal{H}}$. In this paper we study the possible asymptotic behavior of $L_{\mathcal{H}}$. We show any such function is either constant, bounded above and below by a polynomial, or bounded below by an exponential. All results require only reasonable growth rates for ${\epsilon }_2$ (namely polynomial).