On diamond-free subposets of the Boolean lattice: An application of flag algebras

Thumbnail Image
Date
2014-01-01
Authors
Kramer, Lucas
Major Professor
Advisor
Ryan Martin
Committee Member
Journal Title
Journal ISSN
Volume Title
Publisher
Altmetrics
Authors
Research Projects
Organizational Units
Organizational Unit
Journal Issue
Is Version Of
Versions
Series
Department
Mathematics
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).

Comments
Description
Keywords
Citation
Source
Subject Categories
Copyright
Wed Jan 01 00:00:00 UTC 2014