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\section{The Ramsey Numbers}

The course is Probabilistic Combinatorics, being lectured by
\[
  \text{Julian}
  \qquad
  \ub{\text{Sa}}_{}
  \ub{\text{has}}_{}
  \ub{\text{ra}}_{}
  \ub{\text{bu}}_{}
  \ub{\text{dhe}}_{``day''}
\]
We will be studying in particular the Ramsey numbers.
Starting with the simplest to define:
\[
  R(k)
  = \min \{n : \text{every red/blue colouring of $E(K_n)$ contains a
  monochromatic $K_n$}\}
.\]
\begin{example*}
  $R(3) \le 6$: pick a vertex.
  Without loss of generality say it has at least 3 red neighbours.
  If any of these are connected by a red edge, then we get a red triangle by
  using the original vertex.
  If none of them are connected by a red edge, then we get a blue triangle
  between them.
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/8559e6cce42441d5.png}
  \end{center}
  $R(3) > 5$: by considering the following picture.
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/144da32603ae4eb0.png}
  \end{center}
  Thus $R(3) = 6$.
\end{example*}

Main question: how fast does $R(k) \to \infty$ as $k \to \infty$?

We will also study:
\[
  R(l, k)
  = \min \{n : \text{every red/blue colouring of $E(K_n)$ contains either a blue
  $K_l$ or a red $K_k$}\}
.\]
For a fixed graph $H$, we define
\[
  R(H)
  = \min \{n : \text{every red/blue colouring of $E(K_n)$ contains a
  monochromatic $H$}\}
.\]

\begin{example*}
  $R(k) = R(K_k)$.
\end{example*}

For $H$ a fixed graph, we define
\[
  \ex(n, H)
  = \max \{e(G) : \text{$G$ is a graph on $n$ vertices and $G \not\supset H$}\}
.\]

\begin{example*}
  Mantel's Theorem says that
  \[
    \ex(n, K_3)
    = \left\lfloor \frac{n^2}{4} \right\rfloor
  .\]
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/4f2dea872719497e.png}
  \end{center}
\end{example*}

\subsection{Binomial Random Graph}

\begin{fcdefn}[Binomial random graph]
  Given $n \in \Nbb$, $p \in (0, 1)$, the \emph{binomial random graph} $G(n, p)$
  is the probability space defined on all graphs on $n$ vertices, where each
  edge is included independently with probability $p$.
\end{fcdefn}

\subsection{Topics in this course}

\begin{itemize}
  \item First examples: ``first moment method''.
  \item $R(3, k)$:
    \begin{itemize}
      \item deletion method
      \item Lov\'asz local lemma
      \item Semi-random method
      \item Hard core model
      \item The triangle free process
    \end{itemize}
  \item Dependent random choice:
    \begin{itemize}
      \item Ramsey numbers $R(H)$ with $H$ sparse
      \item Sidorenko conjecture
      \item Extremal numbers of bipartite graphs
    \end{itemize}
  \item Pseudo-randomness
    \begin{itemize}
      \item $R(H)$ of bounded-degree graphs
      \item size-ramsey numbers
      \item $R(3, 3, k)$
    \end{itemize}
  \item Szemeredi-Regularity lemma
    \begin{itemize}
      \item Roth's Theorem on 3-term arithmetic progressions in dense sets
      \item Ramsey-Tur\'an
    \end{itemize}
  \item Method of graph containers:
    \begin{itemize}
      \item Counting graphs with no $C_4$
      \item $R(3, 3, k)$
      \item $R(4, k)$
    \end{itemize}
\end{itemize}

\subsection{Brief introduction to $R(k)$}

\begin{fcthm}[Erdos-Szekeres, '35]
  $R(k) \le 4^k$.
\end{fcthm}

\begin{proof}[Proof sketch]
  Let $n \ge 4^k$ and let $\chi$ be a colouring of $K_n$.
  Pick a vertex $v$.
  It must have $\ge \half n$ neighbours connected to it by the same
  colour.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/dfd5487249ff4db3.png}
  \end{center}
  Now ignore everything that was connected using the other colour.
  Pick a new vertex from what remains, and apply the process again:
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/acfcf843128e48ac.png}
  \end{center}
  We continue until either $A$ or $B$ gets to size $k$.
  We basically just want
  \[
    n \left( \half \right)^k \left( \half \right)^k
    \ge 1
  ,\]
  i.e. $n \ge 4^k$.
\end{proof}

How about a lower bound?

\begin{example*}
  The following gives $R(k) \ge (k-1)^2 + 1$.
  Quite a long way from $4^k$!
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/ef5b4df318c64aa7.png}
  \end{center}
\end{example*}

\begin{fcthm}[Erdos, '47]
  $R(k) \ge (1 - o(1)) \frac{k}{\sqrt{2} e} 2^{k/2}$.
\end{fcthm}

\begin{notation*}
  $o(1)$ denotes a quantity that $\to 0$ as $k \to \infty$.
\end{notation*}

\begin{proof}
  Let $n = (1 - o(1)) \frac{k}{\sqrt{2} e} e^{k / 2}$ we choose $\chi$ to be a
  random colouring of $E(K_n)$ where the colour of each edge is chosen red /
  blue uniformly and independently.
  We have 
  \begin{align*}
    \Pbb(\text{$\chi$ has a monochromatic $K_k$})
    &= \Pbb \left( \bigcup_{K \in [n]^{(k)}} \{\text{$K$ is a monochromatic
    clique}\} \right) \\
    &\le 2 {n \choose k} 2^{-{k \choose 2}} \\
    &\le 2 \left( \frac{en}{k} \right)^k 2^{-{k \choose 2}} \\
    &= \left( \frac{en}{k} 2^{\left( \frac{k - 1}{2} \right)} \right) \\
    &< 1
  \end{align*}
  by the choice of $n$.
\end{proof}

Big question: Is there an ``explicit'' construction that gives $R(k) \ge (1 +
\eps)^k$, for some fixed $\eps > 0$?
