  We initiate a study of a relaxed version of the standard Erdos-Renyi random
graph model, where each edge may depend on a few other edges. We call such
graphs "dependent random graphs". Our main result in this direction is a
thorough understanding of the clique number of dependent random graphs. We also
obtain bounds for the chromatic number. Surprisingly, many of the standard
properties of random graphs also hold in this relaxed setting. We show that
with high probability, a dependent random graph will contain a clique of size
$\frac{(1-o(1))\log n}{\log(1/p)}$, and the chromatic number will be at most
$\frac{n \log(1/1-p)}{\log n}$.
  As an application and second main result, we give a new communication
protocol for the k-player Multiparty Pointer Jumping (MPJ_k) problem in the
number-on-the-forehead (NOF) model. Multiparty Pointer Jumping is one of the
canonical NOF communication problems, yet even for three players, its
communication complexity is not well understood. Our protocol for MPJ_3 costs
$O(\frac{n\log\log n}{\log n})$ communication, improving on a bound of Brody
and Chakrabarti [BC08]. We extend our protocol to the non-Boolean pointer
jumping problem $\widehat{MPJ}_k$, achieving an upper bound which is o(n) for
any $k >= 4$ players. This is the first o(n) bound for $\widehat{MPJ}_k$ and
improves on a bound of Damm, Jukna, and Sgall [DJS98] which has stood for
almost twenty years.
