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This volume presents some of the research topics discussed at the
2014-2015 Annual Thematic Program Discrete Structures: Analysis and
Applications at the Institute for Mathematics and its Applications
during Fall 2014, when combinatorics was the focus. Leading experts
have written surveys of research problems, making state of the art
results more conveniently and widely available. The three-part
structure of the volume reflects the three workshops held during
Fall 2014. In the first part, topics on extremal and probabilistic
combinatorics are presented; part two focuses on additive and
analytic combinatorics; and part three presents topics in geometric
and enumerative combinatorics. This book will be of use to those
who research combinatorics directly or apply combinatorial methods
to other fields.
This volume presents some of the research topics discussed at the
2014-2015 Annual Thematic Program Discrete Structures: Analysis and
Applications at the Institute for Mathematics and its Applications
during Fall 2014, when combinatorics was the focus. Leading experts
have written surveys of research problems, making state of the art
results more conveniently and widely available. The three-part
structure of the volume reflects the three workshops held during
Fall 2014. In the first part, topics on extremal and probabilistic
combinatorics are presented; part two focuses on additive and
analytic combinatorics; and part three presents topics in geometric
and enumerative combinatorics. This book will be of use to those
who research combinatorics directly or apply combinatorial methods
to other fields.
This book begins with a gentle introduction to the analytical
aspects of the theory of finite Markov chain mixing times and
quickly ramps up to explain the latest developments in the topic.
Several theorems are revisited and often derived in simpler,
transparent ways, and illustrated with examples. The highlights
include spectral, logarithmic Sobolev techniques, the evolving set
methodology, and issues of nonreversibility. This is a
comprehensive, well-written review of the subject that will be of
interest to researchers and students in computer and mathematical
sciences.
Let $\cal{R}$ be the set of all finite graphs $G$ with the Ramsey
property that every coloring of the edges of $G$ by two colors
yields a monochromatic triangle. In this paper the authors
establish a sharp threshold for random graphs with this property.
Let $G(n, p)$ be the random graph on $n$ vertices with edge
probability $p$. The authors prove that there exists a function
$\widehat c=\widehat c(n)=\Theta(1)$ such that for any $\varepsilon
> 0$, as $n$ tends to infinity, $Pr\left G(n,
(1-\varepsilon)\widehat c/\sqrt{n}) \in \cal{R} \right] \rightarrow
0$ and $Pr \left G(n, (1]\varepsilon)\widehat c/\sqrt{n}) \in
\cal{R}\ \right] \rightarrow 1.$. A crucial tool that is used in
the proof and is of independent interest is a generalization of
Szemeredi's Regularity Lemma to a certain hypergraph setti
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