SOME CUBIC TIME REGULARITY ALGORITHMS FOR TRIPLE SYSTEMS

Szemeredi's regularity lemma guarantees that, for fixed epsilon > 0, every graph G = (V, E) admits an epsilon -regular and t -equitable partition pi (G), where t = O(1). These partitions are constructed by Kohayakawa, Rodl, and Thoma in time O(IVI2). Analogous partitions of k -graphs pi(k) are constructed by Czygrinow and Rodl in time O(IVI2k (- 1)log(5)IVI). For k = 3, we construct these partitions (and others with slightly stronger regularity) in time O(IVI3). We also discuss some applications.