  Chung and Graham began the systematic study of k-uniform hypergraph
quasirandom properties soon after the foundational results of Thomason and
Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in
the early work on k-uniform hypergraph quasirandomness is that properties that
are equivalent for graphs are not equivalent for hypergraphs, and thus
hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past
two decades, there has been an intensive study of these disparate notions of
quasirandomness for hypergraphs, and an open problem that has emerged is to
determine the relationship between them.
  Our main result is to determine the poset of implications between these
quasirandom properties. This answers a recent question of Chung and continues a
project begun by Chung and Graham in their first paper on hypergraph
quasirandomness in the early 1990's.
