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 -homogeneous partitions of size at most 2^2^{^{-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.