\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}
\begin{document}
\module{Ramsey numbers}{2}{20}{2 periods}
\prereq{Basic notions and vocabulary about graphs.}
\object{\begin{itemize}
\item Introduce the concept of a complete graph.
\item Modelize a problem using graphs.
\item Discover the concept of Ramsey numbers.
\end{itemize}}
\mater{\begin{itemize}
\item Beamer.
\end{itemize}}
\modpart{A party problem}{1 period}
The following problem is introduced :
\begin{quote}
You're organising a party. Some of the guests will know each other, while others won't. For the sake of simplicity, we
admit that the relation of knowing each other is symmetric.
What is the least number of guest to invite so that at least $m$ people will mutually know each other, or at least $n$ people will be complete strangers ?
\end{quote}
Some simple examples are studied with the whole class :
\begin{itemize}
\itemb $m=2$ and $n=2$ ;
\itemb $m=3$ and $n=2$ ;
\itemb $m=2$ and $n=3$ ;
\itemb $m=2$ and $n\in\N$ ;
\end{itemize}
Then, students have to stand up and work in groups in the problem with $m=3$ and $n=3$.
If there is enough time, the problem with $m=4$ and $n=3$ can be tackled too.
\modpart{Turning it into a graph problem}{1 period}
The class is asked to modelize this problem with the use of graph.
Hints can be given by the teacher :
\begin{itemize}
\item What will be the vertices of the graph ?
\item How to symbolize the fact the people will know each other or not ?
\end{itemize}
The concepts of complete graph and Ramsey number are introduced, along with the official notations $K_n$ and $R(m,n)$. A quick biography of Ramsey is given.
The proof of the fact that $R(3,3)=6$ is given.
\end{document}