Two problems in extremal combinatorics

Abstract

In this thesis, we focus on two problems in extremal graph theory. Extremal graph theory consists of all problems related to optimizing parameters defined on graphs. The concept of ``editing'' appears in many key results and techniques in extremal graph theory, either as a means to account for error in structural results, or as a quantity to minimize or maximize. A typical problem in spectral extremal graph theory seeks relationships between the extremes of certain graph parameters and the extremes of eigenvalues commonly associated to graphs. The \emph{edit distance problem} asks the following problem: for any fixed ``forbidden'' graph $F$, how many ``edits'' are needed to ensure that any graph on $n$ vertices can be made to contain no induced copies of $F$. If $F$ is a complete graph, then Tur\'{a}n's Theorem, an early fundamental result in extremal graph theory, provides a precise answer. The \emph{edit distance function} plays an essential role in answering this question and relates to the \emph{speed} of a graph hereditary property $\hh$ as well as the $\hh$-chromatic number of a random graph. The main techniques revolve around so-called \emph{colored regularity graphs (CRGs)}. We find an asymptotically almost sure formula for the edit distance function when $F$ is an Erd\H{o}s-R\'{e}nyi random graph whose density lies in $[1-1/\phi, 1/\phi]\approx [0.382¸0.618]$. As an intermediate step, we make several advances on the application of CRGs, such as the introduction and application of \emph{$p$-prohibited CRGs}. %In \emph{spectral graph theory}, we ask: given graph $G$ and some matrix $M$ which may be naturally associated to $G$, what do the eigenvalues of $M$ say about $G$? For any $n$-vertex graph $G$, its adjacency matrix $A = A_G$ is the $\{0,1\}$-valued $n\times n$ matrix whose $(u,v)$ entry indicates whether $uv$ is an edge of $G$. In $1999$, Gregory, Hershkowitz, and Kirkland defined the \emph{(adjacency) spread} of a graph as the difference between the maximum and minimum eigenvalues of its adjacency matrix. In their paper, since cited $68$ times, the authors conjectured that the graph on $n$ vertices which maximizes spread is the join of a complete graph on $\lfloor 2n/3\rfloor$ vertices with an independent set on $\lceil n/3\rceil$ vertices. We prove this claim for all $n$ sufficiently large. As an intermediate step, we prove an analogous result for the eigenvalues of \emph{graphons} (equivalently, kernel operators on symmetric functions $W:[0,1]^2\to [0,1]$).