Pigeonhole and Extremal Arguments · Hard problem

In a group, every person has at least $4$ friends in the group. Take a LONGEST chain of distinct people $P_1, P_2, \ldots, P_m$ in which consecutive people are friends. All friends of $P_m$ must lie on the chain. This guarantees a cycle of friends $P_i, P_{i+1}, \ldots, P_m, P_i$ with at least how many people?