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\begin{fcthm}[Ajtai, Komlós, Szemeredi]
  \label{thm:AKSpath}
  For $0 < \eps < \frac{1}{3}$, if $G \sim G(n, p)$ where $p = \frac{1 +
  \eps}{n}$, then $G \supset P_l$ where $l \ge \eps^5 n$.
\end{fcthm}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/a488be1d9a784673.png}
\end{center}

\begin{fclemma}[]
  \label{lemma:AKSpath}
  Let $G$ be an $n$ vertex graph with \gls{dfsseq} $X_1, X_2, \ldots$ and
  \[
    \sum_{i = 1}^T X_i
    \ge pT - \eps^5 n
  ,\]
  where $T = \eps^k n^2$.
  Then $G \supset P_l$ where $l \ge \eps^5 n$.
\end{fclemma}

\begin{proof}
  If at time $T$ we have $|B| \ge \frac{n}{3}$, then since $|V(P)| \le \eps^5
  n$, assuming for contradiction $P_l \not\subset G$, for all steps.
  Then there must have been some step $\le T$ in the algorithm where $|A| \ge
  \frac{n}{3}$ and $|B| \ge \frac{n}{3}$.
  Then
  \[
    \eps^2 n^2
    = T
    \ge |A| |B|
    \ge \frac{n^2}{9}
  ,\]
  contradiction.

  So we may assume $|B| \le \frac{n}{3}$.
  So we can assume at time $T$ that the algorithm is still running.
  We have
  \[
    \eps^5 n + |B|
    \ge |V(P)| + |B|
    \ge \sum_{i = 1}^T X_i
    = (*)
  \]
  since, each time we see $X_i = 1$ we move a vertex from $A$ into $V(P) \cup
  B$, and it never returns.
  Now
  \[
    (*)
    \ge pT - \eps^5 n
    \ge \frac{(1 + \eps)\eps^3 n^2}{n} - \eps^5 n
    \ge \frac{(1 + \eps^2)T}{n} + \eps^5 n
  .\]
  So combining we have
  \[
    \frac{n}{3}
    \ge |B|
    \ge \frac{(1 + \eps^2) T}{n}
  .\]
  Also note
  \[
    |A|
    = (n - |B| - |V(P)|)
  \]
  so
  \begin{align*}
    T
    &\ge |A||B| \\
    &\ge (n - |B| - \eps^5 n)|B| \\
    &\ge \left( n - \frac{(1 + \eps^2) T}{n} - \eps^5 n \right)
    \frac{(1 + \eps^2)T}{n} \\
    &\ge (1 - 2\eps^3) (1 + \eps^2) T \\
    &> T
  \end{align*}
  contradiction.
\end{proof}

Reminder of Chernoff bound:

\begin{fclemma}[]
  Let $X \sim \Bin(n, p)$.
  Then, for $0 \le t \le pn$, we have
  \[
    \Pbb(|X - \Ebb X| \ge t)
    \le 2\exp \left( - \frac{t^2}{3pn} \right)
  .\]
\end{fclemma}

\begin{proof}[Proof of \cref{thm:AKSpath}]
  Let $G \sim G(n, p)$, where $p = \frac{(1 + \eps)}{n}$, for $0 < \eps <
  \frac{1}{3}$.
  Let $X_1, X_2, X_3, \ldots$ \gls{dfsseq}.
  This is a sequence of iid Bernoulli random variables with probability $p$.
  So set $T = \eps^2 n^2$ and note
  \begin{align*}
    \Pbb \left( \sum_{i = 1}^{T} X_i < pT - \eps^5 n \right)
    &\le \Pbb \left( \left| \sum_{i = 1}^{T} X_i - pT \right| > \eps^5 n \right)
    \\
    &\le 2 \exp \left( - \frac{\eps^{10} n^2}{3(1 + \eps) \eps^3 n} \right) \\
    &\to 0
  \end{align*}
  since $\eps^5 n < pT = (1 + \eps)\eps^3 n$.
  So apply \cref{lemma:AKSpath} to see that with probability $1 - o(1)$, $G
  \supset P_l$, where $l \ge \eps^5 n$.
\end{proof}

\begin{fcthm}[]
  Let $\eps \to 0$.
  Let $w$ be some function with $w \to \infty$.
  Let $G \sim G(n, p)$.
  Then
  \[
    \bigcomp(G)
    = \begin{cases}
      \frac{\log(\eps^3 n)}{\eps^2}
      & \eps < \frac{n^{-\frac{1}{3}}}{w} \\
      \Theta(n^{\frac{2}{3}}
      & \eps = \Theta(n^{-\frac{1}{3}}) \\
      (2 + o(1)) \eps n
      & \eps = w \cdot n^{-\frac{1}{3}}
    \end{cases}
  \]
\end{fcthm}

\newpage
\section{The threshold for a Hamilton cycle}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/efce0a60b4a04c1f.png}
\end{center}

\begin{fcthm}[Pósa]
  If $p \ge \frac{16 \log n}{n}$ and $G \sim G(n, p)$ then $G \supset C_n$ with
  high probability.
\end{fcthm}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/25d581bd59cb44b3.png}
\end{center}

Let $P = v_1 \cdots v_l$ be a path in a graph $G$.
Let $v_i v_l \in G$.
Define the path $Q$ by
\[
  Q
  = v_1 \cdots v_i v_l v_{l - 1} \cdots v_{i + 1}
.\]
We call this the rotation of $P$ through the edge $v_i v_l$.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/8db7e1b8af554f7f.png}
\end{center}

We write for $P, Q$ on the same vertex set $P \sim Q$ if I can obtain $Q$ by
applying a sequence of rotations to $P$.
We define $P \sim P$.

Define
\[
  X(P)
  = \{x \in V(P) : \text{$x$ is the end of a path}\}
.\]
$Q$ and $Q \sim P$.
\[
  \ol{X}(P)
  = X(P) \cup \{v_i : \text{$v_{i - 1}$ or $v_{i + 1} \in X(P)$}\}
.\]

\begin{fclemma}[]
  \label{lemma:XPpathedges}
  Let $G$ be a graph, and let $P \supset G$ be a path.
  Then
  \[
    e(X(P), V(P) \setminus \ol{X}(P))
    = 0
  .\]
\end{fclemma}