Quasi-random hypergraphs revisited

The quasi-random theory for graphs mainly focuses on a large equivalent class of graph properties each of which can be used as a certificate for randomness. For k-graphs (i.e., k-uniform hypergraphs), an analogous quasi-random class contains various equivalent graph properties including the k-discrepancy property (bounding the number of edges in the generalized induced subgraph determined by any given (k − 1)-graph on the same vertex set) as well as the k-deviation property (bounding the occurrences of “octahedron”, a generalization of 4-cycle). In a 1990 paper [1], a weaker notion of l-discrepancy properties for k-graphs was introduced for forming a nested chain of quasi-random classes, but the proof for showing the equivalence of l-discrepancy and l-deviation, for 2 ≤ l < k, contains an error. An additional parameter is needed in the definition of discrepancy, because of the rich and complex structure in hypergraphs. In this note, we introduce the notion of (l, s)-discrepancy for k-graphs and prove that the equivalence of the (k, s)-discrepancy and the k-discrepancy, for 1 ≤ s ≤ k. We remark that this refined notion of discrepancy seems to point to a lattice structure in relating various quasi-random classes for hypergraphs.