In this paper we study the one-way multi-party communication model,
in which every party speaks exactly once in its turn. For every
fixed k, we prove a tight lower bound of
\Omega{n^{1/(k-1)}} on the probabilistic communication
complexity of pointer jumping in a k-layered tree, where the
pointers of the i-th layer reside on the forehead of the i-th
party to speak. The lower bound remains nontrivial even for k = (\log n)^{1/2 - \Omega(1)} parties. Previous to our work a lower bound was
known only for k=3 \cite{BabaiHayesKimmel01},
and in very restricted models for k>3 \cite{DJS98,Cha07}. Our
results have the following consequences to other models and
problems, extending previous work in several directions.

The one-way model is strong enough to capture {\em general} (non
one-way) multi-party protocols of bounded rounds. Thus we generalize
to this multi-party model results on two directions studied in the classical 2-party model (e.g.~\cite{PaS84,NiW93}).
The first is a round hierarchy: We give an exponential separation
between the power of r and 2 r rounds in general
probabilistic k-party protocols, for any fixed k and r. The
second is the relative power of determinism and nondeterminism: We
prove an exponential separation between nondeterministic and
deterministic communication complexity for general k-party
protocols with r rounds, for any fixed k,r.

The pointer jumping function is weak enough to be a special case of the well-studied disjointness function. Thus we obtain a lower bound of
\Omega\rb{n^{1/(k-1)}} on the probabilistic complexity of
k-set disjointness in the one-way model, which was known only
for k=3 parties. Our result also extends a similar lower bound
for the weaker simultaneous model, in which parties simultaneously
send one message to a referee \cite{BPSW05}.

Finally, we infer an exponential separation between the power of
different orders in which parties send messages in the one-way
model, for every fixed k. Previous to our work such a separation
was only known for k=3 \cite{NiW93}.

Our lower bound technique, which handles functions of high
discrepancy, may be of independent interest. It provides a
``party-elimination'' induction, based on a restricted form of a
direct-product result, specific to the pointer jumping function.