, 2015

It is easy to see that, given three different committees, each pair of them can have two people in common only if all three committees share the same two people, for otherwise the people in the committees would contain four people from whom three committees have been formed. As a corollary, for any clique, there are some two people who belong to all the committees in the clique, and these people are unique.
To derive a contradiction, let us consider two cliques $C_1$ and $C_2$ with two committees in common. There are some two people $A$ and $B$ who belong to all the committees in $C_1$. These two people must also belong to the two committees shared by $C_1$ and $C_2$. But then all the committees in $C_2$ must also include $A$ and $B$. Now, we can extend the clique $C_1$ into $C_1 \cup C_2$, which violates the definition of a clique. □

Consider three committees in a clique. As above, each pair of them can have two people in common only if all three of them share the same two people. Therefore, all committees in a clique share the same two people, and the clique with intersection $\{A, B\}$ consists, by maximality, of all possible committees
$$\{A, B, P_1\}, \dots, \{A, B, P_n\}.$$
The clique is thus uniquely determined by the intersection of any two of its elements. □