general Y-channel is considered first, and upper bounds for its sum-capacity are given in section \[UpperBounds\] and lower bounds in sections \[LowerBound1\] and \[LowerBound2\]. The gap between upper and lower bounds is calculated in section \[GapCalculation:General\]. The restricted Y-channel is considered in section \[Section:RYC\] and we summarize in section \[Summary\]. Throughout the paper, we use $x^n$ to denote a sequence of $n$ symbols $(x_1,\dots,x_n)$, we use $C(x)=\frac{1}{2}\log(1+x)$, and $[x]^+=\max\{0,x\}$.

System Model {#Model}
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The Y-channel models a setup where 3 users want to communicate with each other in a bi-directional manner, and this communication is only possible via a relay as shown in Figure \[Fig:Model\]. Each user has an individual message to each other users. Consequently, each user wants to broadcast 2 messages via the relay, and wants to decode 2 messages. We assume that all nodes are full duplex, and that there is an AWGN channel between each node and the relay, where the noise is of zero-mean and unit-variance.

User $j$ has messages $$\begin{aligned}
m_{jk}&\in\mathcal{M}_{jk}\triangleq\{1,\dots,2^{nR_{jk}}\}, \text{and }\\ m_{jl}&\in\mathcal{M}_{jl}\triangleq\{1,\dots,2^{nR_{jl}}\}\end{aligned}$$ to users $k$ and $l$ respectively where $R_{jk},R_{jl}\in\mathbb{R}_+$, for all distinct $j,k,l\in\{1,2,3\}$. The messages of user $j$ are encoded into a sequence $x_j^n$ using an encoder $f_j$, where for $i=1,\dots,n$, 