The quasi-randomness of hypergraph cut properties

Let α1, . . . , αk satisfy ∑ i αi = 1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A1, . . . , Ak of sizes α1n, . . . , αkn, the number of edges intersecting A1, . . . , Ak 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 α1 = · · · = αk = 1/k. This resolves an open problem raised in 1991 by Chung and Graham [J. AMS ’91]. While hypergraphs satisfying the property corresponding to α1 = · · · = αk = 1/k 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.