  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.
