are fulfilled $$\begin{aligned}
R_{12}+R_{13}+R_{31}+R_{32}&\leq C(h_2^2P_r)\\
\label{DFC2}
R_{12}+R_{13}+R_{21}+R_{23}&\leq C(h_3^2P_r).\end{aligned}$$ In order to find the maximum achievable sum rate, we solve $$\begin{aligned}
\label{OptProb}
\text{maximize}\quad &\sum_{j=1}^3\sum_{\substack{k=1\\k\neq j}}R_{jk}\\
\text{subject to}\quad & R_{jk}\geq0 \quad\forall j,k\in\{1,2,3\},\ j\neq k\nonumber\\
& R_{21}+R_{23}+R_{31}+R_{32}\leq C(h_1^2P_r)\nonumber\\
&R_{12}+R_{13}+R_{31}+R_{32}\leq C(h_2^2P_r)\nonumber\\
&R_{12}+R_{13}+R_{21}+R_{23}\leq C(h_3^2P_r).\nonumber\end{aligned}$$ Solving the linear program (\[OptProb\]) keeping (\[Ordering\]) in mind, we obtain (see Appendix \[LPMax\]) $$\begin{aligned}
\label{BCconstraints}
R_\Sigma\leq \min\left\{\sum_{j=2}^3C(h_j^2P_r),\frac{1}{2}\sum_{j=1}^3C(h_j^2P_r)\right\}.\end{aligned}$$

Hence, we obtain the following lower bound for the sum-capacity.

\[LowerBound:DF\] The sum-capacity of the Y-channel satisfies $C_g\geq\underline{C}^I$ where $$\begin{aligned}
\label{DFconstraint}
\underline{C}^I&=\min\left\{C\left(\sum_{j=1}^3h_j^2P\right),\sum_{j=2}^3C(h_j^2P_r),\frac{1}{2}\sum_{j=1}^3C(h_j^2P_r)\right\}.\end{aligned}$$

The maximum achievable sum rate using complete decode and forward is given by the minimum of (\[MACconstraint\]) and (\[BCconstraints\]). Therefore (\[DFconstraint\]) is a lower bound for the sum-capacity.

Lower Bound: Functional Decode and Forward {#LowerBound2}
==========================================

In this section, we describe another achievable scheme that gives us a lower bound for the sum-capacity of the Y-channel. In this scheme, time is divided into frames of 3 time slots each, where in each slot, only 2 users and the relay are active. These blocks will be indexed as block $3f+s$ where $f\in\mathbb{N}$ denotes the frame index and $s\in\{1,2,3\}$ the slot index.

Briefly, in block $3f+s$, the two active users send, say $x_1^n(m_{12}(f))$ and $x_2^n(m_{21}(f))$ to the relay, $m_{12}(f),m_{21}(f)\in\{1,\dots,2^{nR_{12}}\}$. The relay decodes the 