  A network where three users communicate with each other via a relay is
considered. Users do not receive other users' signals via a direct link, and
thus the relay is essential for their communication. Each user is assumed to
have an individual message to be delivered to each other user. Thus, each user
wants to send two messages and to decode two messages. In general, the transmit
signals of different nodes can be dependent since they can depend on previously
received symbols. We call this case the general case. The sum-capacity is
studied, and upper bounds and lower bounds are given. If all nodes have the
same power, the sum-capacity is characterized to within a gap of 5/2 bits or a
factor of 3 for all values of channel coefficients. This gap is also shown to
approach 3/2 bits as the transmit power increases. Moreover, for the symmetric
case with equal channel coefficients, the gap is shown to be less than 1 bit.
The restricted case is also considered where the transmit signal does not
depend on previously received symbols. In this case, the sum-capacity is
characterized to within a gap of 2 bits or a factor of 3 for all values of
channel coefficients, and approaches 1 bit as the transmit power increases.
