  The purpose of this paper is to provide methods for determining the
associated primes of the square of the Alexander dual of the edge ideal for an
m-hypergraph H. We prove a general method for detecting associated primes of
the square of the Alexander dual of the edge ideal based on combinatorial
conditions on the m-hypergraph. Also, we demonstrate a more efficient
combinatorial criterion for detecting the non-existence of non-minimal
associated primes. In investigating 3-hypergraphs, we prove a surprising
extension of the previously discovered results for 2-hypergraphs (simple
graphs). For 2-hypergraphs, associated primes of the square of the Alexander
dual of the edge ideal are either of height 2 or of odd height greater than 2.
However, we prove that in the 3-hypergraph case, there is no such restriction -
or indeed any restriction - on the heights of the associated primes. Further,
we generalize this result to any dimension greater than 3. Specifically, given
any integers m, q, and n with 3\leq m\leq q\leq n, we construct a m-hypergraph
of size n with an associated prime of height q. We further prove that it is
possible to construct connected m-hypergraphs under the same conditions.
