${\mathcal{R}}_1(p)$), where the union is over $p(.)\in{\mathcal{P}}$ (defined in (\[eqn:pmf\_ach\])).

\[thm:ach2\] The union of rate regions given by (\[eqn:ach1\_I\])-(\[eqn:ach1\_III\]) and

[rcl]{} R\_[1]{}&& I(X\_1;Y\_2) \[eqn:ach2\_I\]\
R\_[2]{}&& I(X\_2;Y\_1) \[eqn:ach2\_II\]\
R\_[1]{}+R\_[2]{}+R\_[3]{}&& I(X\_1,X\_2,X\_3;Y\_3) \[eqn:ach2\_III\]

is achievable for the three-user C-IFC (denoted as ${\mathcal{R}}_2(p)$), where the union is over $p(.)\in{\mathcal{P}}$ (defined in (\[eqn:pmf\_ach\])).

Strong Interference Conditions and Capacity Results {#sec:Cap}
===================================================

In this section, we derive two sets of strong interference conditions (*Set1* and *Set2*), under which the regions of Theorem \[thm:ach1\] and Theorem \[thm:ach2\] achieve capacity. First, we give an intuition about deriving the conditions at each receiver:

-   **Strong interference at the cognitive receiver (Rx3):** In *both schemes*, Rx3 jointly decodes $m_1$, $m_2$ and $m_3$. Therefore, it is assumed that $m_1$ and $m_2$ jointly cause strong interference. These conditions are shown in (\[eqn:set1\_IV\]) and the second terms of (\[eqn:set1\_II\]) and (\[eqn:set1\_III\]) for the first scheme. In other words, assuming the above conditions, the joint received signal from Tx1 and Tx2 at Rx3 is strong enough to decode without imposing any rate constraint on $R_{1}$ and $R_{2}$. Similar conditions are also provided for the second scheme in (\[eqn:set2\_IV\]) and the second terms of (\[eqn:set2\_II\]) and (\[eqn:set2\_III\]). Therefore, there is no difference between two schemes 