An algorithmic hypergraph regularity lemma

Szemerédi’s Regularity Lemma [22, 23] is a powerful tool in graph theory. It asserts that all large graphs G admit a bounded partition of E(G), most classes of which are bipartite subgraphs with uniformly distributed edges. The original proof of this result was non-constructive. A constructive proof was given by Alon, Duke, Lefmann, Rödl and Yuster [1], which allows one to efficiently construct a regular partition for any large graph. Szemerédi’s Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl [3] gave one such extension to 3-uniform hypergraphs, and Rödl and Skokan [19] extended this result to k-uniform hypergraphs. W.T. Gowers [4, 5] gave another such extension. Similarly to the graph case, all of these proofs are nonconstructive. We present an efficient algorithmic version of the Hypergraph Regularity Lemma for k-uniform hypergraphs.