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\begin{note*}
  ``Hypergraph'' in these lectures will usually mean $3$-uniform hypergraph.
\end{note*}

\newpage
\section{Hypergraph regularity}

\begin{fcprop}[]
  % Proposition 2.1
  \label{prop:2.1}
  Let $X, Y, Z$ be sets and let $f : X \times Y \times Z \to [-1, 1]$ be a
  function.
  Then the following are equivalent:
  \begin{enumerate}[(i)]
    \item $\Ebb_{x_0, x_1 \in X} \Ebb_{y_0, y_1 \in Y} \Ebb_{z_0, z_1 \in Z}
      \prod_{(i, j, k) \in \{0, 1\}^3} f(x_i, y_j, z_k) \le c_1$.
    \item $|\Ebb_{x \in X, y \in Y, z \in Z} f(x, y, z) u(x, y) v(x, z) w(y, z)|
      \le c_2$ for any $u : X \times Y \to [-1, 1]$, $v : X \times Z \to [-1,
      1]$, $w : Y \times Z \to [-1, 1]$.
    \item For any tripartite graph $G$ on $X \cup Y \cup Z$, the average of $f$
      restricted to $\triangle(G) : \text{triangles in $G$}$ is at most $c_3$ in
      absolute value.
  \end{enumerate}
\end{fcprop}

The objects we count in (i) are \emph{octahedra}:
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/f75b9a3392914b75.png}
\end{center}

\begin{proof}
  Omitted (but the proof is the same as the proof of \cref{prop:1.5}, only with
  more Cauchy Schwarzing).
\end{proof}

\begin{fcdefn}[]
  \glsadjdefn{badhepsquasi}{$\eps$-quasirandom}{hypergraph}%
  Say $f : X \times Y \times Z \to [-1, 1]$ is \emph{$\eps$-quasirandom} if it
  satisfies \cref{prop:2.1}(i) with $c_1 = \eps^8$.
  A $3$-partite $3$-uniform hypergraph $H$ on $X \cup Y \cup Z$ of density
  $\delta$ is \emph{$\eps$-quasirandom} if $f(x, y, z) = \indicator{H}(x, y, z)
  - \delta$ is $\eps$-quasirandom.
\end{fcdefn}

\begin{fcprop}[]
  % Proposition 2.3
  \label{prop:2.3}
  Let $H$be a $4$-partite $3$-uniform hypergraph on vertex set $X \cup Y \cup Z
  \cup W$.
  Suppose $H(X, Y, Z), H(X, Z, W), H(X, Y, W), H(Y, Z, W)$ are all
  \gls{badhepsquasi} of densities $\delta_{XYZ}, \delta_{XZW}, \delta_{XYW},
  \delta_{YZW}$ respectively.
  Then
  \[
    |\Ebb_{x, y, z, w} \indicator{H}(x, y, z) \indicator{H}(x, y, w)
    \indicator{H}(x, z, w) \indicator{H}(y, z, w)
    - \delta_{XYZ} \delta_{XYW} \delta_{XZW} \delta_{YZW}|
    \le 10\eps
  .\]
\end{fcprop}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/d536ad379eb4400c.png}
\end{center}

\begin{proof}
  Omitted (see Example Sheet).
\end{proof}

But this notion is too strong to facilitate a regularity lemma.

\begin{example}
  % Example 2.4
  \label{eg:2.4}
  Let $G$ be a random tripartite graph on $X \cup Y \cup Z$ of edge density
  $\half$.
  Let $H$ be the $3$-uniform hypergraph consisting of all triangles in $G$.
  The edge density of $H$ is $\frac{1}{8}$.
  The proportion of octahedra: $\left( \half \right)^{12}$.
  But, if $H$ were \glsref[badhepsquasi]{quasirandom}, we would expect $\left(
  \half \right)^8$.

  Not only does $H$ fail to be \glsref[badhepsquasi]{quasirandom}, so does any
  induced subhypergraph.
\end{example}

\begin{example}
  % Example 2.5
  \label{eg:2.5}
  Let $G$ be a tripartite graph $X \cup Y \cup Z$ with each part $G(X, Y), G(X,
  Z), G(Y, Z)$ \glsref[epsquasi]{quasirandom}, of edge density $\delta_{XY},
  \delta_{XZ}, \delta_{YZ}$ respectively.
  Let $H$ be a random hypergraph chosen with $\Pbb = \delta$ from the
  $3$-hypergraph formed by $\triangle(G)$.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/b9192991af3c41c0.png}
  \end{center}
  Then the expected number of octahedra in $H$ is
  \[
    (\delta_{XY}\delta_{XZ} \delta_{YZ})^4
    \delta^8
    |X|^2 |Y|^2 |Z|^2
  .\]
\end{example}

\begin{fcdefn}[]
  \glsadjdefn{hepsquasi}{$\eps$-quasirandom}{function or hypergraph}%
  % Definition 2.6
  \label{defn:2.6}
  Let $G$ be a $3$-partite graph on vertex set $X \cup Y \cup Z$, and let $f : X
  \times Y \times Z \to [-1, 1]$ be a function supported on $\triangles(G)$.
  We say $f$ is \emph{$\eps$-quasirandom} relative to $G$ if
  \[
    \Ebb_{\substack{x_0, x_1 \in X \\ y_0, y_1 \in Y \\ z_0, z_1 \in Z}}
    \prod_{(i, j, k) \in \{0, 1\}^3} f(x_i, y_j, z_k)
    \le \eps^8(\delta_{XY} \delta_{XZ} \delta_{YZ})^4
  .\]
  Let $H$ be a $3$-partite $3$-uniform hypergraph on $X \cup Y \cup Z$, and
  suppose $H \subseteq \triangles(G)$, $|H| = \delta|\triangles(G)|$.
  We say $H$ is \emph{$\eps$-quasirandom} relative to $G$ if
  \[
    f(x, y, z)
    = \begin{cases}
      \indicator{H}(x, y, z) - \delta
      & \text{if $(x, y, z) \in \triangles(G)$} \\
      0
      & \text{otherwise}
    \end{cases}
  \]
  is $\eps$-quasirandom relative to $G$.
\end{fcdefn}

\begin{fcprop}[]
  % Proposition 2.7
  \label{prop:2.7}
  Let $G$ be a $4$-partite graph on $X \cup Y \cup Z \cup W$ with all six
  bipartite parts $\eps$-quasirandom.
  Let $f : X \times Y \times Z \to [-1, 1]$, $g : X \times Y \times W
  \to [-1, 1]$, $h : X \times Z \times W \to [-1, 1]$, $k : Y \times Z \times W
  \to [-1, 1]$, all supported on $\triangles(G)$.
  Suppose that $f$ is \hquasi{\eta} relative to $G(X, Y, Z)$.
  Then provided $\eps^{\quarter} \le 2^{-38} (\eta \delta)^8$,
  \[
    |\Ebb_{x, y, z, w} f(x, y, z) g(x, y, w) h(x, z, w) k(y, z, w|
    \le 2\eta \delta
  ,\]
  where 
  \[
    \delta
    \defeq \delta_{XY} \delta_{XZ} \delta_{XW} \delta_{YZ} \delta_{YW}
    \delta_{ZW}
  .\]
\end{fcprop}