3. User 1 also wants to decode two messages, $m_{21}$ from user 2, and $m_{31}$ from user 3. Similarly at users 2 and 3.[]{data-label="Fig:Model"}](Y_Channel.eps){width="0.8\columnwidth"}

In the following sections, we will deal with the sum-capacity of the Y-channel, by deriving upper and lower bounds. Then we bound the gap between the upper and lower bounds. We consider both the general Y-channel where the encoding functions are as given in (\[SourceEncoder\]) whose sum-capacity will be denoted $C_g$, and the restricted Y-channel where the encoding functions are as given in (\[SourceEncoder\_R\]) whose sum-capacity will be denoted $C_r$. Clearly, $C_r\leq C_g$.

General Y-channel: Upper bounds {#UpperBounds}
===============================

We start by considering the general Y-channel, and give sum-capacity upper bounds for this case. One way to obtain upper bounds for the Y-channel is by using the cut-set bounds [@CoverThomas]. If we label the set of nodes in the Y-channel by $\mathcal{S}\triangleq\{U_1,U_2,U_3,R\}$ where $U_j$ denotes user $j$, $j\in\{1,2,3\}$ and $R$ denotes the relay, then the cut-set bounds provide upper bounds on the rate of information flow from a set $\mathcal{T}\subset\mathcal{S}$ to its complement $\mathcal{T}^c$ in $\mathcal{S}$ (see Figure \[Cut\]). The cut-set bounds for this setup yield the following upper bounds.

\[CutSetBounds\] The achievable rates in the 