Date of Award

1-1-2011

Document Type

Campus Access Dissertation

Department

Mathematics

First Advisor

Jerrold R Griggs

Abstract

We study the families of subsets of finite sets that have some special properties. For all families with the properties, we look for the largest sizes of the families. In the thesis, we use a method to bound the size of families of subsets, which is what we call the Lubell function now. The method is a generalization of the idea that Lebell et al. used to show Sperner's Theorem.

On the one hand, we study the Lubell function of families with a forbidden poset(partially ordered set) for many posets and prove that these posets satisfy a conjecture of Griggs and Lu. On the other hand, we give the new proofs of the old results on this type of problems using the Lubell function method, which greatly shorten those proofs.

In addition, we study the maximum value of the Lubell function of families with a forbidden subposet for many posets. This not only helps us to tackle the conjecture of Griggs and Lu but also arises many interesting research problems.

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