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\newpage
\section{Size Ramsey number of a graph}

\begin{fcdefn}[$G$ -> $H$]
  \glssymboldefn{rto}%
  Let $G$, $H$ be graphs.
  We write
  \[
    G \to H
  \]
  to mean every red / blue colouring of $G$ contains a monochromatic $H$.
\end{fcdefn}

\begin{example*}
  $K_6 \rto K_3$.
\end{example*}

\begin{example*}
  $R(k) = \min \{n : K_n \rto K_k\}$.
\end{example*}

\textbf{Question:} How few \emph{edges} can a graph $G$ have with $G \rto H$?

\begin{fcdefn}[Size Ramsey number]
  \glssymboldefn{sizenum}%
  The \emph{size Ramsey number} of $H$ is
  \[
    \hat{r}(H)
    = \min \{e(G) : G \rto H\}
  .\]
\end{fcdefn}

These Ramsey numbers are hard to study in general, so it makes sense to study
some specific cases.
The first case one might think of is studying when $H$ is a clique, but it turns
out that this case is a little bit degenerate:

\begin{fcthm}[]
  $\sizenum(K_k) = {R(k) \choose 2}$.
\end{fcthm}

Similarly to before, we'll study the case where $H$ is a sparse graph.
In fact, we'll focus specifically on paths.

\subsection{The size Ramsey number of the path}

Straightforward bounds:
\[
  2k
  \le \sizenum(P_k)
  \le (ck)^2
.\]
The upper bound is by using the bound on Ramsey numbers of sparse graphs proved
last section (\nameref{thm:grr}).

Surprisingly:

\begin{fcthm}[Beck, '83]
  \label{thm:beck}
  $\sizenum(P_k) \le Ck$, for some constant $C > 0$.
\end{fcthm}

\begin{fcthm}[Beck, '83 (Beck 2)]
  \label{thm:beck2}
  There exists $c, C > 0$, such that the following holds:
  
  Let $p = \frac{C}{n}$ and let $G \sim G(n, p)$.
  Then
  \[
    \Pbb(G \rto P_k)
    = 1 - o(1)
  \]
  where $k = cn$.
\end{fcthm}

\begin{fcdefn}[]
  \glsadjdefn{dpseudor}{$\delta$-pseudorandom}{graph}%
  Let $G$ be a graph on $n$ vertices.
  Say that $G$ is \emph{$\delta$-pseudorandom} if all sets $A, B \subset V(G)$
  which are disjoint and $|A|, |B| \ge 8n$ have $e(A, B) > 0$.
\end{fcdefn}

We will show that any \gls{dpseudor} graph $G$ satisfies $G \rto P_k$ (for
$\delta \le \frac{1}{16}$), and by the following lemma, this will be enough to
deduce \nameref{thm:beck2}.

\begin{fclemma}[]
  \label{lemma:Gnpispseudo}
  Assuming:
  - $\delta > 0$
  Then:
  there exists $C \ge \frac{4}{\delta^2}$ such that if $p = \frac{C}{n}$ and $G
  \sim G(n, p)$ then
  \[
    \Pbb(\text{$G$ is \gls{dpseudor}})
    = 1 - o(1)
  .\]
\end{fclemma}

\begin{proof}
  For $A, B \in [n]^{(\delta n)}$ disjoint,
  \begin{align*}
    \Pbb(e(A, B) = 0)
    &= (1 - p)^{|A||B|} \\
    &\le e^{-p\delta^2 n^2} \\
    &\le e^{-C\delta^2 n}
  \end{align*}
  Then
  \begin{align*}
    \Ebb(\text{\# of $A, B \in [n]^{(\delta n)}$ with $e(A, B) = 0$})
    &\le 2^{2n} \cdot e^{-C\delta^2 n} \\
    &\to 0
  \end{align*}
  So done by Markov.
\end{proof}

\begin{fclemma}[]
  \label{lemma:beckdfs}
  Assuming:
  - $G$ a graph on $n$ vertices
  - $G \not\supset P_k$
  - $a, b \in \Zbb_{\ge 0}$ such that $a + b = n - k$
  Then:
  there exist disjoint $A, B \subset V(G)$ with $|A| = a$, $|B| = b$ and $e(A,
  B) = 0$.
\end{fclemma}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/42f2b556f36a4116.png}
\end{center}

\begin{proof}
  We apply the following ``Depth first search algorithm''.
  We maintain a partition $V(G) = A \cup B \cup P$ where $P$ is a path, and $B$
  we think of as a ``bin''.
  First set $A = V(G)$, $B = P = \emptyset$.
  We now repeat the following until $A = \emptyset$:
  \begin{enumerate}[(1)]
    \item If $P = \emptyset$, then move a vertex from $A$ into $P$.
    \item If $P \neq \emptyset$, let $v$ be the end of the path $P$.
      If $\exists u \in N(v) \cap A$, we move $u$ to the end of $P$.
      If $N(v) \cap A = \emptyset$, we move $v$ into $B$.
  \end{enumerate}
  We note that since $G$ is $P_k$-free, we have $|V(P)| \le k$ and therefore
  \[
    |A \cup B|
    = |A| + |B|
    \ge n - k
  \]
  always.

  Also note that at each step we:
  \begin{itemize}
    \item Either decrease $A$ by $1$; or
    \item We increase $|B|$ by $1$.
  \end{itemize}
  Since $|A| = n$ at the start and $|A| = 0$ at the end, there must be some time
  for which $|A| = a$ and $|B| \ge b$.

  Now since $e(A, B) = 0$ at all times, we are done.
\end{proof}

\begin{proof}[Proof of \nameref{thm:beck2}]
  Let $G$ be a graph on $n$ vertices that is \gls{dpseudor} with $\delta =
  \frac{1}{16}$.
  Let $k = \eps n$, where $\eps > 0$ we choose later.
  We show $G \rto P_k$.
  So assume not.
  Let $G = G_R \cup G_B$ where $G_R \not\supset P_k$, $G_B \not\supset P_k$.
  Apply \cref{lemma:beckdfs} to both $G_R$ and $G_B$ to find $A, B, A', B'
  \subset V(G)$ with $|A| = |B| = |A'| = |B'| = \left( \half - \frac{\eps}{2}
  \right) n$ and such that $A, B$ have no red edges between then and $A', B'$
  have no blue edges between them.
  \begin{center}
    \includegraphics[width=0.3\linewidth]{images/ac4854fb9877499f.png}
  \end{center}
  Note
  \[
    |(A \cup B) \cap A'|
    \ge \left( \half - 2\eps \right) n
  .\]
  Without loss of generality assume
  \[
    |A \cap A'|
    \ge \left( \quarter - \eps \right) n
  .\]
  So
  \[
    |A \cap B'|
    \le |A| - |A \cap A'|
    \le \left( \quarter - 2\eps \right) n
  .\]
  Thus
  \[
    |B \cap B'|
    \ge \left( \quarter - 4 \eps \right) n
  .\]
  So $A \cap A'$ and $B \cap B'$ have no edges between them in $G$, which
  contradicts the \glsref[dpseudor]{$\delta$-pseudorandomness} if $\eps =
  \frac{1}{32}$.
  So $G \rto P_k$.

  So by \cref{lemma:Gnpispseudo} (which said that $G(n, \frac{C}{n})$ is
  \gls{dpseudor} with high probability), we are done.
\end{proof}

\begin{proof}[Proof of \nameref{thm:beck}]
  Enough to prove for $k$ sufficiently large.
  We set $k = \eps n$ as before, $\delta = \frac{1}{16}$ and let $C >
  \frac{4}{\delta^2}$.
  Then $G \sim G(n, p)$ satisfies
  \[
    G \rto P_k
    \quad \text{whp}
  .\]
  Now note
  \[
    \Pbb(e(G) \ge pn^2)
    \le \frac{\Ebb e(G)}{pn^2}
    \le \half
  .\]
  So (at least) $\half - o(1)$ proportion of all such graphs work.
\end{proof}