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\documentclass[12pt,table]{beamer}
 
\usepackage{hyperref}
 
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% For fancy footnotes
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\usepackage{tikz}
 
%\usepackage{preamble}
\usepackage{xspace}
\newcommand{\LLL}{Lov\'asz Local Lemma\xspace}
\begin{document}
 
\title{A Constructive Proof of the Lov\'asz Local Lemma}
\subtitle{Robin Moser, STOC 2009}
\author{Youngtae Youn}
\institute{CSE Dept. Penn State}
\date{September 28, 2009}
 
%%%
\begin{frame}
  \frametitle{\LLL: Lov\'asz and Erd\H{o}s in 1975}
  \begin{itemize}
  \item What if bad events are \alert{not} independent? 
    \begin{itemize}
    \item Bad events are \emph{mostly} independent from one
      another.
    \item Bad event are \alert{not} individually too likely to occur.
    \end{itemize}
    Still, $\Pr[\text{none of the bad events will occur}] > 0$.
  \end{itemize}
 
  \begin{block}{\LLL (Symmetric Case)}
    \begin{enumerate}
    \item $\Pr[bad_i] \leq \emph{p}$ for all $1 \leq i \leq n$.     
    \item Each $bad_i$ depends on other \emph{$d$} bad events.
    \item $e\cdot p\cdot (d+1) \leq 1$ where $\emph{e}=2.7182\cdots$
    \end{enumerate}
    \vspace{-4mm}
    \begin{center}
      Then, $\Pr[\text{none of the bad events will occur}] > 0$.
    \end{center}
  \end{block}
 
  \begin{itemize}
  \item dictionary definition of \emph{local}: affecting or limited to
    part of a whole.
  \item The bound in (3) is \alert{tight}.    
  \item We will prove for the \emph{loose} bound $4pd\leq 1$.
  \end{itemize}
\end{frame}
 
\end{document}

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