The degrees of freedom (DoF) number of the fully connected K-user Gaussian interference channel is known to be K over 2 (see ). In , the DoF for the same channel model was studied while allowing each message to be available at its own transmitter as well as M 1 successive transmitters. In particular, it was shown that the DoF gain through cooperation does not scale with the number of users K for a fixed value of M, i.e., the per user DoF number is 1 over 2. In this work, we relax the cooperation constraint such that each message can be assigned to M transmitters without imposing further constraints on their location. Under the new constraint, we study properties for different message assignments in terms of the gain in the per user DoF number over that achieved without cooperation. In particular, we show that a local cooperation constraint that confines the transmit set of each message within a o(K) radius cannot achieve a per user DoF number that is greater than 1 over 2. Moreover, we show that the same conclusion about the per user DoF number holds for any assignment of messages such that each message cannot be available at more than two transmitters. Finally, for the case where M > 2, we do not know whether a per user DoF number that is greater than 1 over 2 is achievable. However, we identify a candidate class of message assignments that could potentially lead to a positive answer.