Union-Free Families of Subsets

This paper discusses the question of how many non-empty subsets of the set $[n]=\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ we can choose so that no chosen subset is the union of some other chosen subsets. Let $M(n)$ be the maximum number of subsets we can choose. We construct a series of such families, which leads to lower bounds on $M(n)$. We also give upper bounds on $M(n)$. Finally, we propose several conjectures on the tightness of our lower bound for $M(n)$.