Ahlswede-Körner network

The setup
Let $X$ and $Y$ be two sources distributed according to a probability mass function $p_{XY}(\cdot,\cdot)$ on the alphabet $\mathcal{X}\times\mathcal{Y}$. Consider a network with $3$ nodes, labelled nodes $1$, $2$, and $3$. Node $1$ observes a sequence $X_1,X_2,\ldots,X_n$ and node $2$ observes $Y_1, Y_2,\ldots,Y_n$, where the pair $(X_i,Y_i)\sim p_{XY}(\cdot,\cdot)$. Unlike the setup of the Slepian-Wolf Theorem, here, node $3$ wishes to reconstruct only the source sequence $X_1,X_2,\ldots, X_n$.



Source Codes
Let $\mathbf{R}=(R_1,R_2)\in\mathbb{R}^2$. A rate $\mathbf{R}$ blocklength-$n$ code for this setup consists of two encoder maps $f^{(n)}_1$ and $f^{(n)}_2$ with \begin{align} f^{(n)}_1 &: \mathcal{X}^n \rightarrow \{1, 2, \ldots, 2^{nR_1}\} \end{align} and \begin{align} f^{(n)}_2 &: \mathcal{Y}^n \rightarrow \{1, 2, \ldots, 2^{nR_2}\}, \end{align} and a decoder map $g^{(n)}$ with \begin{equation} g^{(n)} : \{1, 2, \ldots, 2^{nR_1}\} \times \{1, 2, \ldots, 2^{nR_2}\} \rightarrow \mathcal{X}^n. \end{equation} The probability of error for this code is given by \begin{equation} P_e^{(n)} = Pr\left\{X^n \neq g\left(f_1\left(X^n\right), f_2\left(Y^n\right)\right)\right\} \end{equation}

Achievable Rates
Ahlswede and Körner showed that there exists a sequence of functions $\{f^{(n)}_1,f^{(n)}_2,g^{(n)}\}_{n=1}^{\infty}$ for which $\lim_{n\to\infty}P_e^{(n)}=0$ if and only if the following inequalities are satisfied:

\begin{align} R_{1} &\geq H\left(X|U\right)\\ R_{2} &\geq I\left(U;Y\right) \end{align} for some random variable $U$ satisfying the Markov chain $U-Y-X$.

Achievability
The two key components to the achievability of the above rates are random binning and joint typicality.

Converse
The key component to the converse is Fano's inequality