29 Let Γ be the graph with vertices corresponding to the 6 people, and edges joining 2 vertices if the people know each other.

Suppose there is a vertex P of degree 3 or more. Let the vertices joined to P be X = {A, B, C, …}. If two of the vertices Q, R ∈ X can be joined, then P, Q, R are ‘mutual acquaintances’. If not, A, B, C, … are ‘mutual strangers’.

Thus we may assume that every vertex has degree ≤ 2. Take any vertex P. Then there are at least 3 vertices not joined to P. If these are all joined to one another then thet are ‘mutual acquaintances’, while if two, say Q, R are not joined then P, Q, R are mutual strangers.

30 Let Γ be the corresponding graph.

Suppose there is a vertex P of degree 4 or more. As in the last question, if 2 of the vertices Q, R joined to P are themselves joined then P, Q, R are ‘mutual acquaintances’. Otherwise the 4 or more vertices joined to P are mutual strangers.

Thus we may assume that every vertex has degree ≤ 3. Take any vertex P. There are at least 6 vertices not joined to P. From the last question, there are either 3 mutual acquaintances among these 6, or else 3 mutual strangers. In the latter case, these 3 vertices together with P give 4 mutual strangers.