  Let a_1,...,a_k satisfy a_1+...+a_k=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 a_1*n,...,a_k*n, the number of edges
intersecting A_1,...,A_k is 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 a_1=...=a_k=1/k. This resolves an open
problem raised in 1991 by Chung and Graham [J. AMS '91].
  While hypergraphs satisfying the property corresponding to a_1=...=a_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.
