Regular slices for hypergraphs

We present a ‘Regular Slice Lemma’ which, given a k-graph G, returns a regular (k−1)-complex J with respect to which G has useful regularity properties. We believe that many arguments in extremal hypergraph theory are made considerably simpler by using this lemma rather than existing forms of the Strong Hypergraph Regularity Lemma, and advocate its use for this reason.