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\begin{fcprop}[]
  % Proposition 1.5
  \label{prop:1.5}
  Let $X, Y$ be sets, and $f : X \times Y \to [-1, 1]$.
  Then the following are equivalent:
  \begin{enumerate}[(i)]
    \item For any pair of sets $X' \subseteq X$, $Y' \subseteq Y$,
      \[
        \Ebb_{x \in X', y \in Y'} f(x, y)
        \le c_1 \frac{|X|}{|X'|} \frac{|Y|}{|Y'|}
      ,\]
      where $\Ebb_{x \in X} = \frac{1}{|X|} \sum_{x \in X}$.
    \item For any two functions $u : X \to [-1, 1]$, $v : Y \to [-1, 1]$,
      \[
        |\Ebb_{x \in X, y \in Y} f(x, y) u(x) v(y)|
        \le c_2
      .\]
    \item $\Ebb_{x, x' \in X} \Ebb_{y, y' \in Y} f(x, y) f(x', y) f(x, y')
      f(x', y') \le c_3$.
  \end{enumerate}
\end{fcprop}

\begin{proof}
  \leavevmode
  \begin{enumerate}[({ii}) $\Rightarrow$ ({iii})]
    \item[(ii) $\Rightarrow$ (i)] Obvious (let $u = \indicator{X'}$, $v =
      \indicator{Y'}$). $c_1 = c_2$.
    \item[(i) $\Rightarrow$ (ii)] Example sheet. $c_2 = \frac{c_1}{12}$.
    \item[(ii) $\Rightarrow$ (iii)] Suppose (iii) is false, i.e.
      \[
        \Ebb_{x, x' \in X} \Ebb_{y, y' \in Y} f(x, y) f(x', y) f(x, y') f(x', y')
        > c_2
      ,\]
      with $c_3 = c_2$.
      We can rewrite as:
      \[
        \Ebb_{x' \in X, y' \in Y}
        \Ebb_{x \in X, y \in Y} f(x, y) \ub{f(x', y)}_{v(y)} \ub{f(x, y') f(x',
        y')}_{u(x)}
        > c_2
      .\]
      So there exist $x' \in X$, $y' \in Y$ such that
      \[
        \Ebb_{x' \in X, y' \in Y}
        \Ebb_{x \in X, y \in Y} f(x, y) v(y) u(x)
        > c_2
      ,\]
      contradiction. $c_3 = c_2$
    \item[(iii) $\Rightarrow$ (ii)]
      \phantom{} \\[-2.7\baselineskip]
      \begin{align*}
        |\Ebb_{x \in X} \Ebb_{y \in Y} f(x, y) u(x) v(y)|^2
        &= |\Ebb_{y \in Y} v(y) \Ebb_{x \in X} f(x, y) u(y)|^2 \\
        &\le \Ebb_{y \in Y} |\Ebb_{x \in X} f(x, y) u(x)|^2 \\
        &= \Ebb_{x \in X} u(x) u(x') \Ebb_{y \in Y} f(x, y) f(x', y)
      \end{align*}
      So
      \begin{align*}
        |\Ebb_{x \in X} \Ebb_{y \in Y} f(x, y) u(x) v(y)|^4
        &\le \Ebb_{x, x' \in X} |\Ebb_{y \in Y} f(x, y) f(x', y)|^2 \\
        &\le c_3
      \end{align*}
      $c_2 = c_3^{\quarter}$
  \end{enumerate}
\end{proof}

\begin{fcdefn}[]
  \glsadjdefn{epsquasi}{$\eps$-quasirandom}{function or graph}%
  % Definition 1.6
  \label{defn:1.6}
  Say $f : X \times Y \to [-1, 1]$ is \emph{$\eps$-quasirandom} if
  \cref{prop:1.5}(iii) holds with $c_3 = \eps^4$.
  Say a graph $G$ on $X \times Y$ of density $\delta > 0$ is
  \emph{$\eps$-quasirandom} if its balanced function $f(x, y) = \indicator{G}(x,
  y) - \delta$ is $\eps$-quasirandom (note that $\Ebb f(x, y) = 0$).
\end{fcdefn}

\begin{fcprop}[Counting Lemma]
  % Proposition 1.7
  \label{prop:1.7}
  Let $G$ be a tripartite graph on vertex sets $X, Y, Z$.
  Suppose $G(X, Y), G(X, Z), G(Y, Z)$ are \gls{epsquasi}, with densities
  $\delta_{XY}, \delta_{XZ}, \delta_{YZ}$ respectively.
  Then
  \[
    |\ub{\Ebb_{x \in X} \Ebb_{y \in Y} \Ebb_{z \in Z} \indicator{G}(x, y)
    \indicator{G}(x, z) \indicator{G}(y, z) - \delta_{XY}\delta_{XZ}
    \delta_{YZ}}_{(*)}|
    \le 4\eps
  .\]
\end{fcprop}

\begin{proof}
  \begin{align*}
    (*)
    &= \Ebb_{x \in X} \Ebb_{y \in Y} \Ebb_{z \in Z} (f_{XY}(x, y) +
    \delta_{XY})(f_{XZ}(x, z) + \delta_{XZ})(f_{YZ}(y, z) + \delta_{YZ})
    - \delta_{XY} \delta_{XZ} \delta_{YZ} \\
    &= \text{seven terms}
  \end{align*}
  For example,
  \[
    \Ebb_{x \in X} \Ebb_{y \in Y} \Ebb_{z \in Z} f_{XY}(x, y) f_{XZ}(x, z)
    f_{YZ}(y, z)
    = \Ebb_{x \in Z} \Ebb_{x \in X, y \in Y} f_{XY}(x, y) u_z(x) v_z(y)
    \le \eps
  ,\]
  by (iii) $\implies$ (ii).
  Similarly for the other terms, and hence we get $(*) \le 7\eps$.

  We can improve it to $(*) \le 4\eps$ by noticing that $3$ of the terms are
  zero (any of the terms of the form $f\delta\delta$).
\end{proof}

\begin{fcdefn}[]
  \glsnoundefn{msd}{mean square density}{graph}%
  % Definition 1.8
  \label{defn:1.8}
  Suppose $X = X_1 \cup \cdots \cup X_n$ and $Y = Y_1 \cup \cdots \cup Y_m$.
  Then the \emph{mean square density} (msd) of a bipartite graph $G$ on $X
  \times Y$ relative to this partition is
  \[
    \sum_{i = 1}^n \sum_{j = 1}^m \frac{|X_i| |Y_j|}{|X||Y|} d(G(X_i, Y_j))^2
  ,\]
  where $d(G(X_i, Y_j))$ is the density of $G(X_i, Y_j)$.
\end{fcdefn}

\begin{example*}
  With respect to the trivial partition, $\msd(G) = d(G(X, Y))^2$.
\end{example*}

\begin{fclemma}[]
  % Lemma 1.8
  \label{lemma:1.8}
  Let $G$ be a bipartite graph on $X \times Y$ of density $\delta > 0$, and
  suppose it fails to be \gls{epsquasi}.
  Then there are partitions $X = X_1 \cup X_2$, $Y = Y_1 \cup Y_2$ such that the
  $\msd$ of $G$ relative to the new partition is $\delta^2 + \left(
  \frac{\eps}{2} \right)^8$.
\end{fclemma}

\begin{proof}
  \cref{prop:1.5}(i) $\Rightarrow$ (iii) provides us with $X_1 \subseteq X$,
  $Y_1 \subseteq Y$ such that
  \[
    \Ebb_{x \in X_1, y \in Y_1} f(x, y)
    > \frac{\eps^4}{12} \frac{|X|}{|X_1|} \frac{|Y|}{|Y_1|}
  .\]
  Let $X_2 = X \setminus X_1$, $Y_2 = Y \setminus Y_1$.

  The $\msd$ relative to the new partition is
  \[
    \sum_{i = 1}^{2} \sum_{j = 1}^{2} \frac{|X_i| |Y_i|}{|X||Y|} d(G(X_i,
    Y_j))^2
  ,\]
  where
  \[
    d(G(X_i, Y_j))
    = \Ebb_{x \in X_i, y \in Y_j} \indicator{G}(x, y)
    = \Ebb_{x \in X_i, y \in Y_j}(f(x, y) + \delta)
    = \delta + \varphi(X_i, Y_j)
  .\]
  Know: $\varphi(X_1, Y_1) > \frac{\eps^4}{12} \frac{|X|}{|X_1|}
  \frac{|Y|}{|Y_1|}$.
  So
  \begin{align*}
    \sum_{i = 1}^{2} \sum_{j = 1}^{2} \frac{|X_i||Y_j|}{|X||Y|} (\delta^2 +
    2\delta\varphi(X_i, Y_j) + \varphi(X_i, Y_j)^2)
    &\ge \delta^2 + 0
    + \sum_{i = 1}^{2} \sum_{j = 1}^{2} \frac{|X_i||Y_j|}{|X||Y|} \varphi(X_i,
    Y_j)^2 \\
    &\ge \delta^2 + \left( \frac{\eps}{2} \right)^8
    \qedhere
  \end{align*}
\end{proof}
