\documentclass[12pt]{article}
\usepackage{amssymb,amsthm}
\usepackage{epsfig}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{construction}[theorem]{Construction}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}

\def\ep{{\varepsilon}}
\def\RR{{\mathbb R}}

\def\proofend
{\ifhmode
\unskip\nobreak\hfill\vrule height5pt width4pt depth2pt\medskip\fi
\ifmmode\eqno{\vrule height5pt width4pt depth2pt}\fi}



\begin{document}

\title{{\bf On a problem of S. Somebody}}
\author{Arthur Author}
\date{October 29, 1999}


\maketitle

\begin{abstract}
Originally we decided not to stick to any strict format but ask the authors to
submit the files either in PLAIN TEX or in LATEX. AMS-LATEX is also
accepted. The reason why AMSTEX is not re\-commended is that 
our final format will be LATEX and AMSTEX files are more difficult to
transform to LATEX.

We thought we put a sample file on our home-page to make the uniformization
easier. This file is not to make your life miserable.

{\bf Suggestions:} Do not use too many definitions and before sending your file to
us check if it contains unnecessary definitions. If yes, delete them, but 
doublecheck if they are really unnecessary. 

If you think your paper is more easily readable if you add figures or graphs
to it, do it. Below we show a way to do it if you can create PostScript files.

{\bf Disclaimer:}
{\it This file has no serious mathematical content. Our aim is just to help you to
create a LATEX file as near to the style we shall use in the Erd\H{o}s volume
as possible.}
\end{abstract}

\section{Notation}
\newcommand{\text}[1]{\quad\mbox{#1}\quad}
\def\beq{\begin{equation}}    \def\eeq{\end{equation}}
\def\beqn{\begin{eqnarray}}   \def\eeqn{\end{eqnarray}}


\begin{quote}
It is worth using LATEX for survey papers, since it is easy to add new parts
to the paper, or new references. Another advantage is that it is easy to add
figures. 
\end{quote}


\section{Introduction}

\begin{theorem}[Weierstrass]\label{Weier}Given a continuous function $f$ on an
interval $[a,b]\subseteq \RR$, this function attains its maximum: there exists
 an $x_0\in [a,b]$ such that
$$f(x_0)\ge f(x)\text{for~every}x\in [a,b].$$
\end{theorem}

\section{Coloring of infinite graphs}

 A nice theorem on coloring of graphs is

\begin{theorem}[de Bruijn-Erd\H{o}s] \label{BruijnErd} If every finite
subgraph of an infinite graph $G$ is $k$-colorable then $G$ is also
$k$-colorable.
\end{theorem}


Now, it is easy to refer to earlier assertions:

Of course, here we cannot see any connection between Theorem \ref{Weier} and
Theorem \ref{BruijnErd}. One connection could be that both they are connected
to compactness.

\begin{definition}
Given a graph $H$, we shall say that it satisfies the density condition with
$p,\varepsilon$ if for every $X\subseteq V(H)$, $|X|>\varepsilon v(H)$ we have
$$(p-\varepsilon){n\choose 2}\le e(H[X])\le (p+\varepsilon){n\choose 2}.$$
\end{definition}

\section{Proofs}

\begin{lemma}[Covering]\label{cover}
If $H_m$ is obtained from $K_m$ by deleting $\le \ep m^2$ edges then one can
find $\ge m(1-\sqrt{2\ep \ell})$ vertex-disjoint copies of $K_\ell$.
\end{lemma}


{\bf Proof.} No proofs will be given here. \proofend

\begin{remark}\rm\footnote{Try out, what happens without changing to ``roman''.}
One may use many ``proofend'' marks. One advantage of the above one is that it
can used easily within formulas, e.g.
$$\int_0^1 x^2\, dx \, .\proofend$$
\end{remark}


\section{How to add figures?}

There are many ways to insert figures into your paper.


\begin{enumerate}

\item Latex has its own means to create figures.

\item Below you will {\em not} see a figure which was created by a new version of
xfig (Linux/Unix) which was able to ``export'' this figure in LATEX form.
(It could be seen on the screen, but did not print out).
\item Then we could decide if we insert this ``picture'' into the file or
use the ``input'' command. However, inserting the ``picture'' generated by
xfig may cause some problems, as we have experienced. So, before using this
LATEX export feature of xfig, please check whether it works. 

\item Another way to insert figures is to use the psfig package or epsfig
package (see the third line of this file).

\begin{center}
\epsfig{file=icosahed.eps,height=8cm,width=8cm}
\end{center}

\end{enumerate}

\begin{thebibliography}{99}

\bibitem{FranklRodlWil} P. Frankl, V. R\"odl and R. M. Wilson: The
number of submatrices of given type in an Hadamard matrix and related
results, Journal of Combinatorics, {\bf 44}(3) (1988) 317-328.

\bibitem{SzemRegu} E. Szemer\'edi: On regular partitions of graphs, Problemes
Combinatoires et Th\'eorie des Graphes (ed. J. Bermond et al.), CNRS Paris,
1978, pp 399-401.

\end{thebibliography}


\end{document}



