Equivalent Conditions for Regularity

Haviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1/2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. Chung and Graham proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this note, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0 < d < 1) and prove that the condition of having asymptotically vanishing discrepancy is equivalent to several other quasi-random properties of H, similar to the ones introduced by Chung and Graham. In particular, we give a proof of the fact that having the correct ‘spectrum’ of the s-vertex subhypergraphs is equivalent to quasi-randomness for any s ≥ 2k. Our work can be viewed as an extension of the results of Chung and Graham to the case of an arbitrary constant valued density. Our methods, however, are based on different ideas.