I have taken the following problem from an introductory text on graph theory. Somehow I am missing something about this problem, as the statement seems wrong to me.
There is a group of $n$ people $A_1, \ldots,A_n$ in which some people know each other and any two people who do not know each other would have common acquaintance. Suppose that $A_1$ and $A_2$ know each other but do not have common acquaintance. Show that the acquaintances of $A_1$ are as many as those of $A_2.$
It is obvious how to model this question in the graph theoretical framework. What I am wondering is why isn't $P_3$ (picking a leaf and a 2-vertex for $A_1$ and $A_2$) a counterexample. The two leaf nodes do not know each other but have common friend, but $A_1$ and $A_2$ do not have the same number of acquaintances.
What am I missing?