Under approval voting, every voter may vote for any number of canditates. To model approval voting, we let a political spectrum be the set of all possible political positions, and let each voter have a subset of the spectrum that they approve, called an approval region. The fraction of all voters who approve the most popular position is the agreement proportion for the society. We consider voting in societies whose political spectrum is modeled by $d$-dimensional space ($\mathbb{R}^d$) with approval regions defined by axis-parallel boxes. For such societies, we first consider a Tur&#aacute;n-type problem, attempting to find the maximum agreement between pairs of voters for a society with a given level of overall agreement. We prove a lower bound on this maximum agreement and find in the literature a proof that the lower bound is optimal. By this result we find that for sufficiently large $n$, any $n$-voter box society in $\mathbb{R}^d$ where at least $\alpha\binom{n}{2}$ pairs of voters agree on some position must have a position contained in $\beta n$ approval regions, where $\alpha = 1-(1-\beta)^2/d$. We also consider an extension of this problem involving projections of approval regions to axes. Finally we consider the question of $(k,m)$-agreeable box societies, where a society is said to be $(k, m)$-agreeable if among every $m$ voters, some $k$ approve a common position. In the $m = 2k - 1$ case, we use methods from graph theory to prove that the agreement proportion is at least $(2d)^{-1}$ for any integer $k \ge 2.$

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics. We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such general purpose bound). We then elaborate on some attempted generalizations/modifications of the Lovász number to undirected hypergraphs that we have tried. It is not currently known whether these attempted generalizations/modifications upper bound the capacity of arbitrary hypergraphs. An important method for proving lower bounds on hypergraph capacity is to exhibit a large independent set in a strong power of the hypergraph. We examine methods for this and show a barrier to attempts to usefully generalize certain of these methods to hypergraphs. We then look at cap sets: independent sets in powers of a certain hypergraph. We examine certain structural properties of them with the hope of finding ones that allow us to prove upper bounds on their size. Finally, we consider two interesting generalizations of capacity and use one of them to formulate several conjectures about connections between cap sets and sunflower-free sets.

In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix $W$ as a subset of $V(G)$ such that all vertices in $W$ have been assigned colors from their respective lists. We give a new, simplified proof where there are a small number of precolored vertices on the same face. We also explore cases where $W=\{u,v\}$ and $G$ has a separating $C_3$ or $C_4$ between $u$ and $v$.