## Graduate Theses and Dissertations

#### Title

Problems in extremal graphs and poset theory

Dissertation

2018

#### Degree Name

Doctor of Philosophy

Mathematics

Mathematics

Ryan R. Martin

#### Abstract

In this dissertation, we present three different research topics and results regarding such topics. We introduce partially ordered sets (posets) and study two types of problems concerning them-- forbidden subposet problems and induced-poset-saturation problems. We conclude by presenting results obtained from studying vertex-identifying codes in graphs.

In studying forbidden subposet problems, we are interested in estimating the maximum size of a family of subsets of the $n$-set avoiding a given subposet. We provide a lower bound for the size of the largest family avoiding the $\mathcal{N}$ poset, which makes use of error-correcting codes. We also provide and upper and lower bound results for a $k$-uniform hypergraph that avoids a \emph{triangle}. Ferrara et al. introduced the concept of studying the minimum size of a family of subsets of the $n$-set avoiding an induced poset, called induced-poset-saturation. In particular, the authors provided a lower bound for the size of an induced-antichain poset and we improve on their lower bound result.

Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. For any nonnegative integer $r$, let $B_r(v)$ denote the ball of radius $r$ around vertex $v\in V$. For a finite graph $G$, an $r$-vertex-identifying code in $G$ is a subset $C\subset V(G)$, with the property that $B_r(u)\cap C\neq B_r(v)\cap C$, for all distinct $u,v\in V(G)$ and $B_r(v)\cap C\neq\emptyset$, for all $v\in V(G)$. We study graphs with large symmetric differences and $(p,\beta)$-jumbled graphs and estimate the minimum size of a vertex-identifying code in each graph.

#### DOI

https://doi.org/10.31274/etd-180810-6112

Shanise Walker

en

application/pdf

62 pages

COinS