Let satisfy and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets of sizes , the number of edges intersecting is (asymptotically) the number one would expect to find in a random k-uniform hypergraph. Can we then infer that H is quasi-random? We show that the answer is negative if and only if . This resolves an open problem raised in 1991 by Chung and Graham [J AMS 4 (1991), 151–196].
While hypergraphs satisfying the property corresponding to are not necessarily quasi-random, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasi-random hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011