\magnification 2000
\parindent 0pt
\parskip 7.5pt
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\hsize 7.2truein
\hoffset -0.2truein
\vsize 9.5truein
\def\u{\vskip -10pt}
\def\v{\vskip 5pt}
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\def\r{\vskip -4pt}

Hypergraph: $(V, E)$, $E \subset {\cal P}(V)$

$X^{(r)}$ = set of all r-tuples of $X$.

A coloring of edges:  $c: X^{(r)} \rightarrow \{red, ~blue\}$

$Y \subset X$ is a red $s$ set if $|Y| = s$ and $c(Y^{(r)}) =$ red.

$R^{(r)}(s, t) = min\{n ~|~ |X| = n$ implies $X^{(r)}$ has a red $s$ set or a blue $t$ set $\}$


Ex:  If $X = \{a, b, c, d\}$ then \hfil \break
$X^{(2)} = \{
\{a, b\}, \{a, c\}, \{a, d\}, \{ b, c\}, \{b, d\}, \{c, d\}
\} = K_4$

\vfill

$R^{(2)}(s, t) = R(s, t)$

$X^{(3)} = \{ \{a, b, c\},\{a, b, d\},\{a, c, d\},\{b, c, d\} \}$

$X^{(4)} = \{\{a, b, c, d\}\}$




\end Ramsey Theory

Pigeonhole Principle:  If you have $n+1$ pigeons in $n$ pigeonholes, then at least one pigeonhole with 
be occupied by 2 or more pigeons.

Example of a Ramsey theorem:  In a group of 6 people, there are either 3 who know each other or 3 who 
are strangers to each other.

van der Waerden't thm:  If $k, p$ are given integers and $W$ is a large enough integer, then any 
partition of the set $\{1,2, 3 ..., W\}$ into $k$ disjoint subsets includes a subset that contains an 
arithmetic progression with $p$ terms.

Fundamental Ramsey Theorems:

Ramsey number = $R(s, t) = min\{n ~|~ $ if the edges of $K_n$ are colored red and blue, then there 
exists either a red $K_s$ or a blue $K_t \}$

$R(3, 3) = 6$.

$R(s, t) = R(t, s)$.

$R(s, 2) = R(2, s) = s$.


Thm 1 (Erdos and Szekeres):  $R(s, t)$ is finite for all $s, t \geq 2$.  If $s > 2$, $t > 2$, then 
$$R(s, t) \leq R(s - 1, t) + R(s, t - 1)$$ and $$R(s, t) \leq \left(\matrix{s + t - 2 \cr s - 
1}\right)$$



\end


List coloring \r

Thm 12:  Let $G$ be a near triangulation with outer cycle $C = x_1x_2...x_k$ such that $L(x_1) = 
\{1\}$, $L(x_2) = \{2\}$, $|L(x_i)| \geq 3$, and $|L(x)| \geq 5$ for all $x \in V(G-C)$.

Cor:  Every planar graph is 5-colorable. \v %%\hrule

\vfill \eject Perfect graphs \s Recall:  $\chi(G) \geq \omega(G) = $ clique number of $G = max\{n ~|~ 
K_n \subset G\}$


$G$ is perfect if $\chi(H) = \omega(H)$ for every induced subgraph $H \subset G$.

Ex:  If $G$ is bipartite, $G$ is perfect.

Ex:  A triangle free graph containing an odd cycle is not perfect.

If $G = (V, E)$, the complement of $G = \overline{G} = (V, V^{(2)} - E)$.

Note:  $\omega(\overline{G}) = \alpha(G)$ independence number of $G$ = maximal size of an independent 
set of vertices.

Thm 15:  The complement of a bipartite graph is perfect.

Recall Cor 3.10:  If $G$ bipartite, then $\alpha(G) =$ minimal number of edges and vertices covering 
all vertices (where $e = xy$ covers $x$ and $x$ covers $x$).

%%Thm 16:  If $G$ bipartite, then the line graph $H = L(G)$ and its complement %%$\overline{H}$ are perfect.

%%Thm 17:  Comparability graphs and their complements are perfect.

Thm 18:  A graph $G$ is perfect iff for every induced subgraph $H \subset G$, there is an independent set of vertices $I$ such that $\omega(H - I) < \omega(I)$

I.e., $G$ is perfect iff every induced subgraph $H$, there exists an independent set of vertices $I$ such that $I \cap D \not= \emptyset$ for every clique $D$ of maximal order $\omega(G)$. 

(recall, a clique = maximal complete subgraph).

Thm 19:  A graph, $H$, obtained from a perfect graph, $G$ by replacing its vertices with perfect graphs, $G_i$ is perfect  ($v(H) = \cup V(G_i)$, $E(H) = [\cup E(G_i)] \cup \{v_iv_j ~|~ v_i \in G_i, v_j \in G_j$ such that the vertices in $G$ corresponding to $G_i, G_j$ are adjacent $\}$).

Thm 20:  The complement of a perfect graph is perfect.





\end