#### Degree Type

Dissertation

#### Date of Award

2016

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### Major

Mathematics

#### First Advisor

Ryan R. Martin

#### Abstract

Let P(n) denote the set of all subsets of {1,...,n} and let P(n,p) be the set obtained from P(n) by selecting elements independently at random with probability p. The Boolean lattice is a partially ordered set, or poset, consisting of the elements of P(n), partially ordered by set inclusion. A basic question in extremal poset theory asks the following: Given a poset P, how big is the largest family of sets in the Boolean lattice which does not contain the structure P as a subposet? The following random analogue of this question is also of interest: Given a poset P, how big is the largest family of sets in P(n,p) which does not contain the structure P as a subposet? In this thesis, we present new proofs for a collection of deterministic extremal subposet problems. We also discuss a new technique called the Hypergraph Container Method in depth and use it to prove a random version of De Bonis and Katona's (r+1)-fork-free theorem.

#### Copyright Owner

Kirsten Ann Hogenson

#### Copyright Date

2016

#### Language

en

#### File Format

application/pdf

#### File Size

60 pages

#### Recommended Citation

Hogenson, Kirsten Ann, "Random and deterministic versions of extremal poset problems" (2016). *Graduate Theses and Dissertations*. 14971.

https://lib.dr.iastate.edu/etd/14971