Hypergraph regularity and quasi-randomness

Thomason and Chung, Graham, and Wilson were the first to systematically study quasi-random graphs and hypergraphs, and proved that several properties of random graphs imply each other in a deterministic sense. Their concepts of quasi-randomness match the notion of e-regularity from the earlier Szemeredi regularity lemma. In contrast, there exists no "natural" hypergraph regularity lemma matching the notions of quasi-random hypergraphs considered by those authors. We study several notions of quasi-randomness for 3-uniform hypergraphs which correspond to the regularity lemmas of Frankl and Rodl, Gowers and Haxell, Nagle and Rodl. We establish an equivalence among the three notions of regularity of these lemmas. Since the regularity lemma of Haxell et al. is algorithmic, we obtain algorithmic versions of the lemmas of Frankl-Rodl (a special case thereof) and Gowers as corollaries. As a further corollary, we obtain that the special case of the Frankl-Rodl lemma (which we can make algorithmic) admits a corresponding counting lemma. (This corollary follows by the equivalences and that the regularity lemma of Gowers or that of Haxell et al. admits a counting lemma.)