Extending the Szemer?edi Regularity Lemma for graphs, P. Frankl and the third author [11] established a 3-graph Regularity Lemma guaranteeing that all large triple systems admit partitions of their edge sets into constantlymany classes where most classes consist of regularly distributed edges. Many applications of this lemma require a companion Counting Lemma [26] allowing one to estimate the number of copies of K_k ^{(3)} in a "dense and regular" environment created by the 3-graph Regularity Lemma. Combined applications of these lemmas are known as the 3- graph Regularity Method. In this paper, we provide an algorithmic version of the 3-graph Regularity Lemma which, as we show, is compatible with a Counting Lemma. We also discuss some applications.

For general k-uniform hypergraphs, Regularity and Counting Lemmas were recently established by Gowers [16] and by Nagle, R?odl, Schacht, and Skokan [27, 35]. We believe the arguments here provide a basis toward a general algorithmic hypergraph regularity method.