  We study three capacity problems in the mobile telephone model, a network
abstraction that models the peer-to-peer communication capabilities implemented
in most commodity smartphone operating systems. The capacity of a network
expresses how much sustained throughput can be maintained for a set of
communication demands, and is therefore a fundamental bound on the usefulness
of a network. Because of this importance, wireless network capacity has been
active area of research for the last two decades.
  The three capacity problems that we study differ in the structure of the
communication demands. The first problem is pairwise capacity, where the
demands are (source, destination) pairs. Pairwise capacity is one of the most
classical definitions, as it was analyzed in the seminal paper of Gupta and
Kumar on wireless network capacity. The second problem we study is broadcast
capacity, in which a single source must deliver packets to all other nodes in
the network. Finally, we turn our attention to all-to-all capacity, in which
all nodes must deliver packets to all other nodes. In all three of these
problems we characterize the optimal achievable throughput for any given
network, and design algorithms which asymptotically match this performance. We
also study these problems in networks generated randomly by a process
introduced by Gupta and Kumar, and fully characterize their achievable
throughput.
  Interestingly, the techniques that we develop for all-to-all capacity also
allow us to design a one-shot gossip algorithm that runs within a
polylogarithmic factor of optimal in every graph. This largely resolves an open
question from previous work on the one-shot gossip problem in this model.
