ELMO 2022 Problem 5

Let $n\ge 3$ be a positive integer. There are $n^3$ users on a social media network called Everyone Likes Meeting Online (ELMO), and some pairs of these users are buddies. A set of at least three ELMO users forms an ELMOClub if and only if all pairs of members of the set are buddies. It is known that among every $n$ users, some three form an ELMOclub. Prove that there is an ELMOclub with five members.

Luke Robitaille