Abstract
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.
Highlights
Let F be a finite family of finite sets with |F| 2
We say that F is union-closed if, for any A, B ∈ F, we have A ∪ B ∈ F
Supposing that F consists of subsets of P(n), define the degree of x ∈ [n] as d(x) := |{K ∈ F | x ∈ K}|
Summary
Let F be a finite family of finite sets with |F| 2. Knowledge spaces are union-closed families containing the empty set that are used to model the knowledge of learners in various academic fields of study [5, 10, 12] Such families that are well-graded (known as learning spaces) have been effectively used in computerized tutoring systems. In what follows we will study the properties of well-graded families and how they relate to the union-closed sets conjecture.
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