University of Illinois Urbana-Champaign

On some problems in extremal, probabilistic and enumerative combinatorics

Wagner, Zsolt Adam

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WAGNER-DISSERTATION-2018.pdf

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https://hdl.handle.net/2142/100954

Description

Title
On some problems in extremal, probabilistic and enumerative combinatorics
Author(s)
Wagner, Zsolt Adam
Issue Date
2018-04-11
Director of Research (if dissertation) or Advisor (if thesis)
Balogh, József
Doctoral Committee Chair(s)
Kostochka, Alexandr V.
Committee Member(s)
  • Tserunyan, Anush
  • Lavrov, Mikhail
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2018-09-04T20:26:59Z
Keyword(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}s--Rademacher 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 Burosch--Demetrovics--Katona--Kleitman--Sapozhenko 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 two-colored, 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 $F(k) \ge 2^{2^{k-1}/k}$ for all $k\in \N$. Fox--Grinshpun--Pach 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}s-Szekeres, which states that any sequence of $n$ numbers contains a monotone subsequence of length at least $\sqrt{n}$.
Graduation Semester
2018-05
Type of Resource
text
Permalink
http://hdl.handle.net/2142/100954
Copyright and License Information
Copyright 2018 Zsolt Wagner

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