SCHWEITZER 1978 Problem 1

Let $\mathcal{H}$ be a family of finite subsets of an infinite set $X$ such that every finite subset of $X$ can be represented as the union of two disjoint sets from $\mathcal{H}$ . Prove that for every positive integer $k$ there is a subset of $X$ that can be represented in at least $k$ different ways as the union of two disjoint sets from $\mathcal{H}$ .

P. Erdos