ACO The ACO Seminar (2010-2011)

May 3, 1:00pm, Wean 8220
Asaf Shapira, Georgia Tech
The Quasi-Randomness of Hypergraph Cut Properties


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 numberof 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.

Joint work with Raphy Yuster

Back to the ACO home page