SCHWEITZER 2018 Problem 2

A family $\mathcal{F}$ of sets is called really neat if for any $A,B\in \mathcal{F}$ , there is a set $C\in \mathcal{F}$ such that $A\cup B = A\cup C=B\cup C$ . Let $$f(n)=\min \left\{ \max_{A\in \mathcal{F}} |A| \colon \mathcal{F} \text{ is really neat and } |\cup \mathcal{F}| =n\right\} .$$ Prove that the sequence $f(n)/n$ converges and find its limit.