The quasi-randomness of hypergraph cut properties

Let α ₁,...,α _k satisfy ∑ _iα _i=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A ₁,...,A _k of sizes α ₁n,...,α _kn, the number of edges intersecting A ₁,...,A _k 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 α ₁= ... =α _k=1/k. 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 α ₁= ... =α _k=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.