Union-Free Families of Subsets

TL;DR

This paper investigates the maximum size of families of subsets of [n] where no subset is the union of others, providing bounds and conjectures on the exact maximum.

Contribution

It introduces new constructions for union-free families and establishes bounds on their maximum size, advancing understanding of their combinatorial properties.

Findings

Constructed new union-free families of subsets.

Provided lower bounds on the maximum size M(n).

Established upper bounds and proposed conjectures.

Abstract

This paper discusses the question of how many non-empty subsets of the set $[n] = \{ 1, 2, ..., n\}$ 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 $M(n)$.