Growth of regular partitions 1: Improved bounds for small slicewise VC-dimension

This is Part 1 in a series of papers about sizes of regular partitions of $3$-uniform hypergraphs. Previous work of the author and Wolf, and independently Chernikov and Towsner, showed that $3$-uniform hypergraphs of small slicewise VC-dimension admit homogeneous partitions. The results of Chernikov and Towsner did not produce explicit bounds, while the work of the author and Wolf relied on a strong version of hypergraph regularity, and consequently produced a Wowzer type bound on the size of the partition. This paper gives a new proof of this result, yielding $\e$-homogeneous partitions of size at most $2^{2^{\epsilon^{-K}}}$, where $K$ is a constant depending on the slicewise VC-dimension. This result is a crucial ingredient in Part 2 of the series, which investigates the growth of weak regular partitions in hereditary properties of $3$-uniform hypergraphs.