Probabilistic and extremal behavior in graphs and matrices
Author(s)McKinley, Gweneth(Gweneth Ann)
Massachusetts Institute of Technology. Department of Mathematics.
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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].
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020Cataloged from the official PDF of thesis.Includes bibliographical references (pages 77-82).
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology