Degree Type

Dissertation

Date of Award

2014

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Ryan Martin

Abstract

The Boolean lattice of dimension two, also known as the diamond, consists of four distinct elements with the following property: A B;C D. The 3-crown consists of six distinct elements with the following property: A a subset of B, D and B a subset of C E and A a subset of C, F. A P-free family in the n-dimensional Boolean lattice is a subposet such that no collection of elements form the poset P. Note that the posets are not induced and may contain additional relations. There is a diamond-free family in the n-dimensional Boolean lattice of size (2+o(1))*( n choose n/2).

In this dissertation, we prove that any diamond-free family in the n-dimensional Boolean lattice has size at most (2:25 + o(1))

*( n choose n/2). Furthermore, we show that the so-called Lubell function of a diamond-free family in the n-dimensional Boolean lattice which contains the empty set is at most 2:25 + o(1), which is asymptotically best possible.

There is a 3-crown-free family in the n-dimensional Boolean lattice of size n choose n/2. In this dissertation, we prove that any 3-crown-free family in the n-dimensional Boolean lattice has

size at most (2(3)&half - 2)*( n choose n/2).

Copyright Owner

Lucas Kramer

Language

en

File Format

application/pdf

File Size

71 pages

Included in

Mathematics Commons

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