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 $|\mathcal{F}| = O(m+n)$?

The terminology comes from addition chain, but instead of addition we have set unions.