Well-graded families and the union-closed sets conjecture (Q2309235)

Summary: The union-closed sets conjecture states that if a finite family of sets \(\mathcal{F}\) is union-closed, then there must be some element contained in at least half of the sets of \(\mathcal{F} \). In this work we study the relationship between the union-closed sets conjecture and union-closed families that have the property of being well-graded. In doing so, we show how the density and other properties are affected by the extra structure contained in well-graded families, and we also give several conditions under which well-graded families satisfy the union-closed sets conjecture.