#### 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

#### Copyright Date

2014

#### Language

en

#### File Format

application/pdf

#### File Size

71 pages

#### Recommended Citation

Kramer, Lucas, "On diamond-free subposets of the Boolean lattice: An application of flag algebras" (2014). *Graduate Theses and Dissertations*. 13894.

https://lib.dr.iastate.edu/etd/13894