# Size of the smallest union chain containing a family

A family of sets $$\mathcal{F}$$ is a union chain if each set in $$\mathcal{F}$$ of size at least $$2$$ is the union of two other sets in $$\mathcal{F}$$. That is, $$X\in \mathcal{F}$$ and $$|X|\geq 2$$, then $$X=Y\cup Z$$ for some $$Y,Z\in \mathcal{F}$$ and $$Y,Z\neq X$$.

Let $$\mathcal{A}$$ be a family of sets. Let $$|\mathcal{A}|=n$$ and $$\left|\bigcup_{A\in \mathcal{A}} A\right|=m$$.

What are some upper bounds for $$|\mathcal{F}|$$ in terms of $$n$$ and $$m$$, where $$\mathcal{F}$$ is the smallest union chain that contains $$\mathcal{A}$$? Is $$|\mathcal{F}| = O(m+n)$$?