independent messages, each of which is intended to one of the other users. Thus each node wants to broadcast 2 messages to the other nodes, and wants to decode 2 other messages. A MIMO variant of this model was considered in [@LeeLim], where a transmission scheme was proposed, and its corresponding achievable degrees of freedom were calculated. It was referred to as the “Y-channel".

We consider the single antenna Gaussian case, where all nodes are full-duplex, and derive upper bounds for the sum-capacity of this channel. We distinguish between two cases: a general Y-channel, and a restricted Y-channel. In the general case, the transmit signals of the users can depend on the previously received symbols, while in the restricted case it can not. In addition to the cut-set bounds, new bounds are derived that are shown to be tighter than the cut-set bounds at moderate to high transmit power.

In [@OngKellettJohnson], the so-called “functional decode and forward" scheme was used as an achievable scheme for the multi-way relay channel. However, in [@OngKellettJohnson], the case where each user has only one message to be delivered to all other users was considered. This is different from our model, where each user has 