Friends and enemies

Pigeonhole at $P$. $P$ has $4$ relations: at least $2$ friends (WLOG) $A,B$.

Among $A,B$. If $A,B$ are friends, then $P,A,B$ are $3$ mutual friends. If $A,B$ are strangers, then... we only get $2$ mutual strangers among $A,B$, not $3$.

Correction. This works for $6$ people, not $5$. For $5$, the statement is false: color $K_5$ without a monochromatic triangle.

Conclusion. For $6$ people, we get $3$ mutual friends or $3$ mutual strangers (Ramsey $R(3,3)=6$). For $5$, it can fail.