  We study the multiparty communication complexity of high dimensional
permutations, in the Number On the Forehead (NOF) model. This model is due to
Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the
Exactly-n problem where three players receive integer inputs and need to decide
if their inputs sum to a given integer $n$. There is a considerable body of
literature dealing with the same problem, where $(\mathbb{N},+)$ is replaced by
some other abelian group. Our work can be viewed as a far-reaching extension of
this line of work.
  We show that the known lower bounds for that group-theoretic problem apply to
all high dimensional permutations. We introduce new proof techniques that
appeal to recent advances in Additive Combinatorics and Ramsey theory. We
reveal new and unexpected connections between the NOF communication complexity
of high dimensional permutations and a variety of well known and thoroughly
studied problems in combinatorics.
  Previous protocols for Exactly-n all rely on the construction of large sets
of integers without a 3-term arithmetic progression. No direct algorithmic
protocol was previously known for the problem, and we provide the first such
algorithm. This suggests new ways to significantly improve the CFL protocol.
  Many new open questions are presented throughout.
