Probabilistic and extremal behavior in graphs and matrices

Abstract

This thesis deals with several related questions in probabilistic and extremal graph theory and discrete random matrix theory. First, for any bipartite graph H containing a cycle, we prove an upper bound of [mathematical equation] on the number of labeled H-free graphs on n vertices, given only a fairly natural assumption on the growth rate of. Bounds of the form [mathematical equation] have been proven only for relatively few special graphs H, often with considerable difficulty, and our result unifies all previously known special cases. Next, we give a variety of bounds on the clique numbers of random graphs arising from the theory of graphons. A graphon is a symmetric measurable function [mathematical equation], and each graphon gives rise naturally to a random graph distribution, denoted G(n, W ), that can be viewed as a generalization of the Erdős-Ré́nyi random graph.

Recently, Doležal, Hladký, and Máthé gave an asymptotic formula of order log n for the clique number of G(n, W ) when W is bounded away from 0 and 1. We show that if W is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of G(n, W) will be [theta]([square root n]) almost surely. We also give a family of examples with clique number [theta](n[superscript alpha]) for any [alpha] [element symbol] (0, 1) , and some conditions under which the clique number of G(n, W ) will be [omicron]([square root]n), [lower case omega]([square root]n), or [upper case omega]([superscript alpha]) for [alpha] [element symbol] (0, 1). Finally, for an nxm matrix M of independent Rademacher (±1) random variables, it is well known that if n </- m, then M is of full rank with high probability; we show that this property is resilient to adversarial changes to M.

More precisely, if m >/- n + n[superscript 1-[epsilon]/6], then even after changing the sign of (1 - [epsilon])m/2 entries, M is still of full rank with high probability. This is asymptotically best possible, as one can easily make any two rows proportional with at most m/2 changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu in [Vu08].