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This paper considers two simple wireless network configurations - the multiple access relay channel (MARC) and the multiple relay channel (MRC) - in which the links making up the network time-share the medium and the assumed loss mechanisms are memoryless packet erasures. The capacity region of the MARC and the capacity of the MRC are derived as functions of the link erasure rates. This is done assuming (i) optimal sharing of bandwidth among the transmitters, and (ii) perfect knowledge at the destination of erasure patterns on all the links. Moreover, it is shown that easily-implemented capacity-approaching codes for the binary erasure channel, such as LDPC or Tornado codes, can be used to attain any achievable rate(s). Finally, these capacity results are unchanged in the presence of feedback of erasure location information to all nodes. For the erasure MARC, the results imply a simple strategy, viz., that the relay should help only those sources that have a weaker direct channel to the destination than the relay itself - regardless of the quality of the source-to-relay channels. For the erasure MRC, the solution has a more complex, inductive structure: the participation of a relay r in the optimal strategy depends on the best throughput that can be achieved using only those relays with a better link to the destination than r.