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Title:  On some problems in extremal, probabilistic and enumerative combinatorics 
Author(s):  Wagner, Zsolt Adam 
Director of Research:  Balogh, Jozsef 
Doctoral Committee Chair(s):  Kostochka, Alexandr V 
Doctoral Committee Member(s):  Tserunyan, Anush; Lavrov, Mikhail 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  extremal combinatorics
probabilistic combinatorics enumerative combinatorics 
Abstract:  This is a study of a small selection of problems from various areas of Combinatorics and Graph Theory, a fast developing field that provides a diverse spectrum of powerful tools with numerous applications to computer science, optimization theory and economics. In this thesis, we focus on extremal, probabilistic and enumerative problems in this field. A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family $\F\subseteq \P(n)$ that does not contain a $2$chain $F_1\subsetneq F_2$. Erd\H{o}s later extended this result and determined the largest family not containing a $k$chain $F_1\subsetneq \ldots \subsetneq F_k$. Erd\H{o}s and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result. In Chapter 2 we answer their question for all families of size at most $(1\eps)2^n$, provided $n$ is sufficiently larger compared to $k$ and $\eps$. The result of Chapter 2 is an example of a supersaturation, or Erd\H{o}sRademacher type result, which seeks to answer how many forbidden objects must appear in a set whose size is larger than the corresponding result. These supersaturation results are a key ingredient to a very recently discovered proof method, called the Container method. Chapters 3 and 4 show various examples of this method in action. In Chapter 3 we, among others, give tight bounds on the logarithm of the number of $t$error correcting codes and illustrate how the Container method can be used to prove random analoges of classical extremal results. In Chapter 4 we solve a conjecture of BuroschDemetrovicsKatonaKleitmanSapozhenko about estimating the number of families in $\{0,1\}^n$ which do not contain two sets and their union. In Chapter 5 we improve an old result of Erd\H{o}s and Spencer. Folkman's theorem asserts that for each $k \in \N$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are twocolored, there exists a set $A \subset [n]$ of size $k$ with the property that all the sums of the form $\sum_{x \in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same color. In 1989, Erd\H{o}s and Spencer showed that $F(k) \ge 2^{ck^2/ \log k}$, where $c >0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \ge 2^{2^{k1}/k}$ for all $k\in \N$. FoxGrinshpunPach showed that every $3$coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $\Omega\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. In Chapter 6 we show, amongst others, that every such coloring contains a $2$colored subgraph with chromatic number at least $n^{2/3}$, and this is best possible. As a direct corollary of our result we obtain a generalisation of the celebrated theorem of Erd\H{o}sSzekeres, which states that any sequence of $n$ numbers contains a monotone subsequence of length at least $\sqrt{n}$. 
Issue Date:  20180411 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/100954 
Rights Information:  Copyright 2018 Zsolt Wagner 
Date Available in IDEALS:  20180904 
Date Deposited:  201805 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois