  Consider the "Number in Hand" multiparty communication complexity model,
where k players holding inputs x_1,...,x_k in {0,1}^n communicate to compute
the value f(x_1,...,x_k) of a function f known to all of them. The main lower
bound technique for the communication complexity of such problems is that of
partition arguments: partition the k players into two disjoint sets of players
and find a lower bound for the induced two-party communication complexity
problem.
  In this paper, we study the power of partition arguments. Our two main
results are very different in nature: (i) For randomized communication
complexity, we show that partition arguments may yield bounds that are
exponentially far from the true communication complexity. Specifically, we
prove that there exists a 3-argument function f whose communication complexity
is Omega(n), while partition arguments can only yield an Omega(log n) lower
bound. The same holds for nondeterministic communication complexity. (ii) For
deterministic communication complexity, we prove that finding significant gaps
between the true communication complexity and the best lower bound that can be
obtained via partition arguments, would imply progress on a generalized version
of the "log-rank conjecture" in communication complexity.
  We conclude with two results on the multiparty "fooling set technique",
another method for obtaining communication complexity lower bounds.
