Growth of regular partitions 3: strong regularity and the vertex partition

This is Part 3 in a series of papers about the growth of regular partitions in $3$-uniform hypergraphs. We consider here the strong regularity for $3$-uniform hypergraphs developed by Frankl, Gowers, Kohayakawa, Nagle, R\"{o}dl, Skokan, and Schacht. This type of regular decomposition comes with two components, a partition of vertices, and a partition of pairs of vertices. We define two corresponding growth functions associated to a hereditary property H of $3$-uniform hypergraphs: TH which measures the size of the vertex component, and LH which measures the size of the pairs component. In this paper, we give an almost complete description of the possible growth rates for TH: constant, polynomial, between single and double exponential, or wowzer. The only existing lower bound constructions for this type of hypergraph regularity were due to Moshkovitz and Shapira, who constructed examples requiring a wowzer-type lower bound on the vertex component in a weaker type of hypergraph regularity. The results of this paper rely crucially on the fact that a slightly simpler construction can be used to produce a wowzer type lower bound for TH. The key ingredient in this simpler example is a lower bound construction for strong graph regularity due to Conlon and Fox.