\documentclass[12pt,letterpaper]{amsart}

\usepackage[leqno]{amsmath}
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\usepackage{amsthm}
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\newcommand{\ZZn}[1]{\ZZ/{#1}\ZZ}
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\newcommand{\zzn}[1]{\zz/{#1}\zz}
\newcommand{\qq}{\mathbb Q}
\newcommand{\calM}{\mathcal M}
\newcommand{\dd}{{\rm d}}
\newcommand{\sm}{\setminus} 


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%These commands deal with theorem-like environments (i.e., italic)
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}

%These deal with definition-like environments (i.e., non-italic)
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

%This numbers equations by section
\numberwithin{equation}{section}



\begin{document}


\title{Title Here}
\author{Your Name \and The Other Name}
\date{\today}

\begin{abstract}
Summarize what your paper is about in one or two paragraphs. Be concise.
\end{abstract}


\maketitle







\section{Introduction}\label{intro}


In this file, edit the information in 
\verb1\title{}1 and \verb1\author{}1.
Do {\it not} change or add typesetting commands to 
the top of the file, which 
affect the size of the output.

Write your paper between  
\verb1\section{Introduction}\label{intro}1 and \verb1\end{document}1.
Hint: if you need any \latex{} construction, an equation for example, 
google it. 

In this section you should put an introduction.  
Tell us what your topic is about, roughly, 
and what you are going to do with it. 


\section{Section 2}\label{sec2}

Make the title of this section a little bit more descriptive than 
the banal ``Section 2.'' 

\begin{definition}
Include formal definitions of all concepts you use.
\end{definition}

\begin{lemma}
A lemma is a small result that helps in proving your big theorem(s).
\end{lemma}

\begin{proof}
And of course, it requires a proof.
\end{proof}

\begin{theorem}
Ah, a larger result. 
\end{theorem}

\begin{proof}
And its proof.
\end{proof}

\begin{corollary}
A corollary is something that follows trivially from the preceeding theorem. It is often a special case with special significance. It usually does not require proof, but sometimes a very short proof is needed to see the connection.
\end{corollary}

This is how to include a figure. Make the figure however you want, and convert it to jpg or pdf format. Then include as follows. \verb10.5\textwidth1 means that the picture is scaled until it is half as wide as the text. 

\begin{figure}[ht]
\label{fig1}
\centerline{\includegraphics[width=0.5\textwidth]{staircase.pdf}}
\caption{This is an example of something counted by the Catalan numbers}
\end{figure}

\section{Section 3}\label{sec3}

What follows is the text of the self-study document, so you can see how it works.

A partially ordered set, or poset, is a set $P$ with a binary relation $\prec$ which is reflexive,
anti-symmetric and transitive. Two elements $a,b\in P$ are comparable if $a\prec b$ or $b\prec a$, otherwise they are incomparable. A chain is a subset $C$ of $P$ so that any two elements of $C$ are comparable. A chain $C$ is maximal if there is no other chain that contains $C$ as a proper subset. An antichain is a subset $A$ of $P$ so that any two elements of $A$ are comparable. An antichain $A$ is maximal if there is no other antichain that contains $A$ as a proper subset.

A maximal element of $P$ is an element $a$ so that, for any $b\in P$, $a\prec b\Rightarrow a=b$. A minimal element is defined similarly. 

\begin{theorem}[Dilworth] Let $P$ be a poset. The minimum number $m$ of disjoint chains which together contain all elements of $P$ is equal to the maximum number $M$ of elements in an antichain of $P$.
\end{theorem}

\begin{proof} Since an antichain and a chain can intersect in at most one element, we have that $m\leq M$. To prove the other part, use induction on $|P|$. If $|P|=0$ there is nothing to prove. Let $C$ be a maximal chain in $P$. If every antichain in $P\setminus C$ contains at most $M-1$ elements, we are done. So assume that $\{ \alpha_1,\dots, \alpha_M\}$ is an antichain in $P\setminus C$. Define $S^-=\{ x\in P: \exists i, x\prec \alpha_i\}$, and define $S^+$ analogously. Since $C$ is maximal, the largest element in $C$ is not in $S^-$ and hence $|S^-|<|P|$ and by the induction hypothesis, the theorem holds for $S^-$. Hence $S^-$ is the union of $M$ disjoint chains. Moreover, each of these chains has exactly one of the elements $\alpha_i$ as its maximal element. Similarly, $S^+$ is the union of $M$ disjoint chains, each of which has exactly  one of the elements $\alpha_i$ as its minimal element. Combining the chains in $S^-$ and $S^+$ that contain the same $\alpha_i$, we obtain $M$ disjoint chains whose union is $P$.
\end{proof}

\begin{theorem}   [Minsky] Let $P$ be a partially ordered set. If $P$ possesses no chain of $m+1$ elements, then $P$ is the union of $m$ antichains.
\end{theorem}
\begin{proof} Induction on $m$. If $m=1$, then all elements of $P$ are incomparable, and $P$ is itself an antichain. Let $m\geq 2$ and assume the theorem is true for $m-1$. Let $M$ be the set of maximal elements of $P$. Clearly, $M$ is an antichain. Let $C$ be any maximal chain in $P$. Then $C$ must contain an element of $M$. Therefore, $P\setminus M$ possesses no chain of $m$ elements. By the induction hypothesis, $P\setminus M$ is the union of $m-1$ antichains. This proves the theorem.
\end{proof}

\begin{theorem}[Sperner's theorem] If $A_1,\dots ,A_m$ are subsets of $[n]$ so that no two sets $A_i$ are subsets of one another, then $m\leq {n\choose \lfloor n/2\rfloor}$.
\end{theorem}

To prove this, consider the poset of subsets of $[n]$ and the relation $\subseteq$. See Jukna, Theorem 8.3.

\begin{theorem}[Erd\"{o}s-Ko-Rado] 
\label{EKR} If $A_1,\dots ,A_m$ are $k$-subsets of $[n]$, $k\leq \lfloor n/2\rfloor$, so that no two sets $A_i$ are disjoint. Then $m\leq {n-1\choose k-1}$.
\end{theorem}

Such a collection of sets is sometimes called an intersecting family. See  Sections 7.1 and 7.2, Jukna.

The following is a folklore result. If the edges of the complete graph $K_7$ are coloured red and blue, then there must be a red or a blue triangle. In general, the Ramsey number $R(r,k)$ is the smallest integer $n$ so that, if the edges of $K_n$ are coloured with $r$ colours, there is a always a monochromatic $K_k$. Ramsey's theorem says that this number is well-defined, i.e.~there always exists such an integer $n$. Read more about Ramsely numbers in Cameron, Section 10.1--4. 



\section{Section 4}\label{sec4}

If you really need it, here is a fourth section.  The labels allow you to refer to a previous section, or theorem. For example, the previous section is Section \ref{sec3}. The Erd\H{o}s-Ko-Rado theorem is Theorem \ref{EKR}. 

You can generate as many sections as you need. 


\appendix

\section{Appendix stuff}\label{app1sec}

Perhaps in the main part of your text there are some 
long and boring calculations 
whose appearance would interrupt the flow of ideas, or there is 
something you want to 
describe briefly but not get bogged down by further in the main 
text of your paper.  If 
further information 
is nevertheless worthwhile, consider placing it in an appendix.  
Here the numbering of equations and theorems gets an A attached to 
reflect the location:

\begin{theorem}
The function $\sin x$ is the unique solution of the differential equation
\begin{equation}\label{sineqn}
\frac{\dd^2y}{\dd{x}^2} + y = 0
\end{equation}
satisfying the initial conditions $y(0) = 0$ and $y'(0) = 1$.
\end{theorem}



\section{More appendix stuff}

It is doubtful that a $7\frac{1}{2}$ page paper should have a second appendix, 
but here is one so you can see it.  All labels get slapped with a B. 

\begin{conjecture}
Papers of this length are unlikely to have a second appendix.
\end{conjecture}

\begin{proof}
We argue by contradiction.  For instance, consider this paper as a base case. 
It has a second appendix.  This is a contrapositive, or is it a converse? 
Well, this proof makes no sense, but it does seem weird 
anyway to be proving a conjecture 
right after stating it.  Better luck next time.
\end{proof}


\begin{thebibliography}{5}

\bibitem{gt}
B. H. Gross and J. T. Tate, Commentary on algebra, pp.~335--336 in: 
``A Century of Mathematics in America, Part II,'' Amer. Math. Soc., 
Providence, 1989.

\bibitem{irros}
K. Ireland and M. Rosen, ``A Classical Introduction to Modern 
Number Theory,'' 2nd ed., Springer-Verlag, New York, 1990.

\bibitem{unabomber}
T. J. Kaczynski, Another proof of Wedderburn's theorem, 
{\it Amer. Math. Monthly} {\bf 71} (1964), 652--653.


\bibitem{knus}
M-A. Knus, A. Merkurjev, M. Rost, J-P. Tignol, 
``The Book of Involutions,'' Amer. Math. Society, Providence, 1998.

\bibitem{roquette}
P. Roquette, Class field theory in characteristic $p$, its origin 
and development, pp.~549--631 in: ``Class field theory -- its centenary 
and prospect,'' Math. Soc. Japan, Tokyo, 2001.

\bibitem{sloane}
N. J. A. Sloane, On-line Encyclopedia of Integer Sequences, 
{\tt http://www.research.att.com/$\sim$njas/ 
sequences/}.

\end{thebibliography}

\end{document}








