For 3-graphs of bounded VC2 dimension, an (ε,ψ)-regular partition exists with twr(twr(poly(1/ε))) vertex parts, improving the generic wowzer bound to tower type.
Terry, An improved bound for regular decompositions of 3-uniform hypergraphs of bounded VC2-dimension, Model Theory 2 (2023), 325–356
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Regularity for hypergraphs with bounded VC$_2$ dimension
For 3-graphs of bounded VC2 dimension, an (ε,ψ)-regular partition exists with twr(twr(poly(1/ε))) vertex parts, improving the generic wowzer bound to tower type.