A storage system requiring file redundancy and on-line repairability can be represented as a Steiner system, a combinatorial design with the property that every $t$-subset of its points occurs in exactly one of its blocks. Under this representation, files are the points and storage units are the blocks of the Steiner system, or vice-versa. Often, the popularities of the files of such storage systems run the gamut, with some files receiving hardly any attention, and others receiving most of it. For such systems, minimizing the difference in the collective popularity between any two storage units is nontrivial; this is the access balancing problem. With regard to the representative Steiner system, the access balancing problem in its simplest form amounts to constructing either a point or block labelling: an assignment of a set of integer labels (popularity ranks) to the Steiner system's point set or block set, respectively, requiring of the former assignment that the sums of the labelled points of any two blocks differ as little as possible and of the latter that the sums of the labels assigned to the containing blocks of any two distinct points differ as little as possible. The central aim of this dissertation is to supply point and block labellings for Steiner systems of block size greater than three, for which up to this point no attempt has been made. Four major results are given in this connection. First, motivated by the close connection between the size of the independent sets of a Steiner system and the quality of its labellings, a Steiner triple system of any admissible order is constructed with a pair of disjoint independent sets of maximum cardinality. Second, the spectrum of resolvable Bose triple systems is determined in order to label some Steiner 2-designs with block size four. Third, several kinds of independent sets are used to point-label Steiner 2-designs with block size four. Finally, optimal and close to optimal block labellings are given for an infinite class of 1-rotational resolvable Steiner 2-designs with arbitrarily large block size by exploiting their underlying group-theoretic properties.

A $k$-list assignment for a graph $G=(V, E)$ is a function $L$ that assigns a $k$-set $L(v)$ of "available colors" to each vertex $v \in V$. A $d$-defective, $m$-fold, $L$-coloring is a function $\phi$ that assigns an $m$-subset $\phi(v) \subseteq L(v)$ to each vertex $v$ so that each color class $V_{i}=\{v \in V:$ $i \in \phi(v)\}$ induces a subgraph of $G$ with maximum degree at most $d$. An edge $xy$ is an $i$-flaw of $\phi$ if $i\in \phi(x) \cap \phi(y)$. An online list-coloring algorithm $\mathcal{A}$ works on a known graph $G$ and an unknown $k$-list assignment $L$ to produce a coloring $\phi$ as follows. At step $r$ the set of vertices $v$ with $r \in L(v)$ is revealed to $\mathcal{A}$. For each vertex $v$, $\mathcal{A}$ must decide irrevocably whether to add $r$ to $\phi(v)$. The online choice number $\pt_{m}^{d}(G)$ of $G$ is the least $k$ for which some such algorithm produces a $d$-defective, $m$-fold, $L$-coloring $\phi$ of $G$ for all $k$-list assignments $L$. Online list coloring was introduced independently by Uwe Schauz and Xuding Zhu. It was known that if $G$ is planar then $\pt_{1}^{0}(G) \leq 5$ and $\pt_{1}^{1}(G) \leq 4$ are sharp bounds; here it is proved that $\pt_{1}^{3}(G) \leq 3$ is sharp, but there is a planar graph $H$ with $\pt_{1}^{2}(H)\ge 4$. Zhu conjectured that for some integer $m$, every planar graph $G$ satisfies $\pt_{m}^{0}(G) \leq 5 m-1$, and even that this is true for $m=2$. This dissertation proves that $\pt_{2}^{1}(G) \leq 9$, so the conjecture is "nearly" true, and the proof extends to $\pt_{m}^{1}(G) \leq\left\lceil\frac{9}{2} m\right\rceil$. Using Alon's Combinatorial Nullstellensatz, this is strengthened by showing that $G$ contains a linear forest $(V, F)$ such that there is an online algorithm that witnesses $\mathrm{pt}_{2}^{1}(G) \leq 9$ while producing a coloring whose flaws are in $F$, and such that no edge is an $i$-flaw and a $j$-flaw for distinct colors $i$ and $j$.

Any permutation in the finite symmetric group can be written as a product of simple transpositions $s_i = (i~i+1)$. For a fixed permutation $\sigma \in \mathfrak{S}_n$ the products of minimal length are called reduced decompositions or reduced words, and the collection of all such reduced words is denoted $R(\sigma)$. Any reduced word of $\sigma$ can be transformed into any other by a sequence of commutation moves or long braid moves. One area of interest in these sets are the congruence classes defined by using only braid moves or only commutation moves. This document will present work towards a conjectured relationship between the number of reduced words and the number of braid classes. The set $R(\sigma)$ can be drawn as a graph, $G(\sigma)$, where the vertices are the reduced words, and the edges denote the presence of a commutation or braid move between the words. This paper will present brand new work on subgraph structures in $G(\sigma)$, as well as new formulas to count the number of braid edges and commutation edges in $G(\sigma)$. The permutation $\sigma$ covers $\tau$ in the weak order poset if the length of $\tau$ is one less than the length of $\sigma$, and there exists a simple transposition $s_i$ such that $\sigma = \tau s_i$. This paper will cover new work on the relationships between the size of $R(\sigma)$ and $R(\tau)$, and how this creates a new method of writing reduced decompositions of $\sigma$ as products of permutations $\alpha$ and $\beta$, where both $\alpha$ and $\beta$ have a length greater than one. Finally, this thesis will also discuss how these results help relate the number of reduced words and the number of braid classes in certain cases.

This dissertation consists of three papers about opinion dynamics. The first paper is in collaboration with Prof. Lanchier while the other two papers are individual works. Two models are introduced and studied analytically: the Deffuant model and the Hegselmann-Krause~(HK) model. The main difference between the two models is that the Deffuant dynamics consists of pairwise interactions whereas the HK dynamics consists of group interactions. Translated into graph, each vertex stands for an agent in both models. In the Deffuant model, two graphs are combined: the social graph and the opinion graph. The social graph is assumed to be a general finite connected graph where each edge is interpreted as a social link, such as a friendship relationship, between two agents. At each time step, two social neighbors are randomly selected and interact if and only if their opinion distance does not exceed some confidence threshold, which results in the neighbors' opinions getting closer to each other. The main result about the Deffuant model is the derivation of a positive lower bound for the probability of consensus that is independent of the size and topology of the social graph but depends on the confidence threshold, the choice of the opinion space and the initial distribution. For the HK model, agent~$i$ updates its opinion~$x_i$ by taking the average opinion of its neighbors, defined as the set of agents with opinion at most~$\epsilon$ apart from~$x_i$. Here,~$\epsilon > 0$ is a confidence threshold. There are two types of HK models: the synchronous and the asynchronous HK models. In the former, all the agents update their opinion simultaneously at each time step, whereas in the latter, only one agent is selected uniformly at random to update its opinion at each time step. The mixed model is a variant of the HK model in which each agent can choose its degree of stubbornness and mix its opinion with the average opinion of its neighbors. The main results of this dissertation about HK models show conditions under which the asymptotic stability holds or a consensus can be achieved, and give a positive lower bound for the probability of consensus and, in the one-dimensional case, an upper bound for the probability of consensus. I demonstrate the bounds for the probability of consensus on a unit cube and a unit interval.

The chromatic number $\chi(G)$ of a graph $G=(V,E)$ is the minimum

number of colors needed to color $V(G)$ such that no adjacent vertices

receive the same color. The coloring number $\col(G)$ of a graph

$G$ is the minimum number $k$ such that there exists a linear ordering

of $V(G)$ for which each vertex has at most $k-1$ backward neighbors.

It is well known that the coloring number is an upper bound for the

chromatic number. The weak $r$-coloring number $\wcol_{r}(G)$ is

a generalization of the coloring number, and it was first introduced

by Kierstead and Yang \cite{77}. The weak $r$-coloring number $\wcol_{r}(G)$

is the minimum integer $k$ such that for some linear ordering $L$

of $V(G)$ each vertex $v$ can reach at most $k-1$ other smaller

vertices $u$ (with respect to $L$) with a path of length at most

$r$ and $u$ is the smallest vertex in the path. This dissertation proves that $\wcol_{2}(G)\le23$ for every planar graph $G$.

The exact distance-$3$ graph $G^{[\natural3]}$ of a graph $G=(V,E)$

is a graph with $V$ as its set of vertices, and $xy\in E(G^{[\natural3]})$

if and only if the distance between $x$ and $y$ in $G$ is $3$.

This dissertation improves the best known upper bound of the

chromatic number of the exact distance-$3$ graphs $G^{[\natural3]}$

of planar graphs $G$, which is $105$, to $95$. It also improves

the best known lower bound, which is $7$, to $9$.

A class of graphs is nowhere dense if for every $r\ge 1$ there exists $t\ge 1$ such that no graph in the class contains a topological minor of the complete graph $K_t$ where every edge is subdivided at most $r$ times. This dissertation gives a new characterization of nowhere dense classes using generalized notions of the domination number.

The construction of many families of combinatorial objects remains a challenging problem. A t-restriction is an array where a predicate is satisfied for every t columns; an example is a perfect hash family (PHF). The composition of a PHF and any t-restriction satisfying predicate P yields another t-restriction also satisfying P with more columns than the original t-restriction had. This thesis concerns three approaches in determining the smallest size of PHFs.



Firstly, hash families in which the associated property is satisfied at least some number lambda times are considered, called higher-index, which guarantees redundancy when constructing t-restrictions. Some direct and optimal constructions of hash families of higher index are given. A new recursive construction is established that generalizes previous results and generates higher-index PHFs with more columns. Probabilistic methods are employed to obtain an upper bound on the optimal size of higher-index PHFs when the number of columns is large. A new deterministic algorithm is developed that generates such PHFs meeting this bound, and computational results are reported.



Secondly, a restriction on the structure of PHFs is introduced, called fractal, a method from Blackburn. His method is extended in several ways; from homogeneous hash families (every row has the same number of symbols) to heterogeneous ones; and to distributing hash families, a relaxation of the predicate for PHFs. Recursive constructions with fractal hash families as ingredients are given, and improve upon on the best-known sizes of many PHFs.



Thirdly, a method of Colbourn and Lanus is extended in which they horizontally copied a given hash family and greedily applied transformations to each copy. Transformations of existential t-restrictions are introduced, which allow for the method to be applicable to any t-restriction having structure like those of hash families. A genetic algorithm is employed for finding the "best" such transformations. Computational results of the GA are reported using PHFs, as the number of transformations permitted is large compared to the number of symbols. Finally, an analysis is given of what trade-offs exist between computation time and the number of t-sets left not satisfying the predicate.

Van der Waerden’s Theorem asserts that for any two positive integers k and r, one may find an integer w=w(k,r) known as the Van der Waerden Number such that for every r-coloring of the integers from 1 to w there exists a monochromatic arithmetic progression of length k. This groundbreaking theorem in combinatorics has greatly impacted the field of discrete math for decades. However, it is quite difficult to find the exact values of w. As such, it would be worth more of our time to try and bound such a value, both from below and above, in order to restrict the possible values of the Van der Waerden Numbers. In this thesis we will endeavor to bound such a number; in addition to proving Van der Waerden’s Theorem, we will discuss the unique functions that bound the Van der Waerden Numbers.

In the ordering game on a graph G, Alice and Bob take turns placing the vertices of G into a linear ordering. The score of the game is the maximum number of neighbors that any vertex has before it in the ordering. Alice's goal in the ordering game is to minimize the score, while Bob's goal is to maximize it. The game coloring number is the least score that Alice can always guarantee in the ordering game, no matter how Bob plays. This paper examines what happens to the game coloring number if Alice or Bob skip turns on the ordering game. Preliminary definitions and examples are provided to give context to the ordering game. These include a polynomial time algorithm to compute the coloring number, a non-competitive version of the game coloring number. The notion of the preordered game is introduced as the primary tool of the paper, along with its naturally defined preordered game coloring number. To emphasize the complex relationship between the coloring number and the preordered game coloring number, a non-polynomial time strategy is given to Alice and Bob that yields the preordered game coloring number on any graph. Additionally, the preordered game coloring number is shown to be monotonic, one of the most useful properties for turn-skipping. Techniques developed throughout the paper are then used to determine that Alice cannot reduce the score and Bob cannot improve the score by skipping any number of their respective turns. This paper can hopefully be used as a stepping stone towards bounding the score on graphs when vertices are removed, as well as in deciphering further properties of the asymmetric marking game.

Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs.

Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.

In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.

Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.

The uncrossing partially ordered set $P_n$ is defined on the set of matchings on $2n$ points on a circle represented with wires. The order relation is $\tau'\leq \tau$ in $P_n$ if and only if $\tau'$ is obtained by resolving a crossing of $\tau$. %This partial order has been studied by Alman-Lian-Tran, Huang-Wen-Xie, Kenyon, and Lam. %The posets $P_n$ emerged from studies of circular planar electrical networks. Circular planar electrical networks are finite weighted undirected graphs embedded into a disk, with boundary vertices and interior vertices. By Curtis-Ingerman-Morrow and de Verdi\`ere-Gitler-Vertigan, the electrical networks can be encoded with response matrices. By Lam the space of response matrices for electrical networks has a cell structure, and this cell structure can be described by the uncrossing partial orders. %Lam proves that the posets can be identified with dual Bruhat order on affine permutations of type $(n,2n)$. Using this identification, Lam proves the poset $\hat{P}_n$, the uncrossing poset $P_n$ with a unique minimum element $\hat{0}$ adjoined, is Eulerian. This thesis consists of two sets of results: (1) flag enumeration in intervals in the uncrossing poset $P_n$ and (2) cyclic sieving phenomenon on the set $P_n$.

I identify elements in $P_n$ with affine permutations of type $(0,2n)$. %This identification enables us to explicitly describe the elements in $P_n$ with the elements in $\mathcal{MP}_n$.

Using this identification, I adapt a technique in Reading for finding recursions for the cd-indices of intervals in Bruhat order of Coxeter groups to the uncrossing poset $P_n$. As a result, I produce recursions for the cd-indices of intervals in the uncrossing poset $P_n$. I also obtain a recursion for the ab-indices of intervals in the poset $\hat{P}_n$, the poset $P_n$ with a unique minimum $\hat0$ adjoined. %We define an induced subposet $\mathcal{MP}_n$ of the affine permutations under Bruhat order.

Reiner-Stanton-White defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action on a finite set and a polynomial. Sagan observed the CSP on the set of non-crossing matchings with the $q$-Catalan polynomial. Bowling-Liang presented similar results on the set of $k$-crossing matchings for $1\leq k \leq 3$. In this dissertation, I focus on the set of all matchings on $[2n]:=\{1,2,\dots,2n\}$. I find the number of matchings fixed by $\frac{2\pi}{d}$ rotations for $d|2n$. I then find the polynomial $X_n(q)$ such that the set of matchings together with $X_n(q)$ and the cyclic group of order $2n$ exhibits the CSP.