IMC 2016 Problem 4

Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties: (i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements; (ii) for any two sets $A, B\in \mathcal{F}$ , their union $A\cup B$ also belongs to $\mathcal{F}$ . Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.

(Proposed by Fedor Petrov, St. Petersburg State University)