Well-Graded Families and the Union-Closed Sets Conjecture

  • Jeffrey Matayoshi


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.

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