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% 12/02/2026 10AM

\begin{fcdefn}[]
  \glsnoundefn{chain}{chain}{chains}%
  \glsnoundefn{chainquasi}{quasirandom}{}%
  % Definition 2.14
  \label{defn:2.14}
  An \emph{$r$-partite chain} is a $(G, H)$, where $G$ is an $r$-partite graph,
  $H$ an $r$-partite hypergraph on the same vertex set as $G$ and $H \subseteq
  \triangles(G)$.
  Let $\psi(\eta, \delta)$ be a polynomial in $\eta$ and $\delta$ which
  vanishes when either is $0$.

  A $4$-partite chain $(G, H)$ is \emph{$(\eta, \psi)$-quasirandom} if
  \begin{itemize}
    \item All four $3$-partite parts of $H$ are \hquasirel{\eta} $G$
    \item all six bipartite parts of $G$ are $\psi(\eta, \delta)$-quasirandom,
      where $\delta$ is the product of the densities of the bipartite parts of
      $G$.
      Shall use this with $\psi(\eta, \delta) = (2^{-38} \eta^8 8^8)^{32}$.
  \end{itemize}
\end{fcdefn}

\begin{remark*}
  We call it a \emph{chain} because when generalised to $4$-uniform, $5$-uniform
  etc, we would use $(G, H, K, \ldots)$ rather than $(G, H)$.
\end{remark*}

\begin{fcdefn}[]
  \glsnoundefn{acd}{associated chain decomposition}{associated chain
  decompositions}%
  % Definition 2.15
  \label{defn:2.15}
  Let $X_1, X_2, X_3, X_4$ be four sets.
  Then a decomposition of the complete $4$-partite graph $K(X_1 \cup X_2 \cup
  X_3 \cup X_4)$ consists of the following:
  \begin{itemize}
    \item For each $X_i$ a partition into subsets $X_i^{(1)} \cup \cdots \cup
      X_i^{(n_i)}$.
    \item For each $K(X_i \cup X_j)$, a partition into subgraphs
      \[
        G_{ij}^{(1)} \cup \cdots G_{ij}^{(m_{ij})}
      .\]
  \end{itemize}

  A typical graph $G$ in this decomposition will have vertex sets $X_i^{(r)},
  X_j^{(s)}, X_k^{(t)}$ and edge set
  \[
    G_{ij}^{(u)}(X_i^{(r)} \cup X_j^{(s)})
    \cup G_{ij}^{(v)}(X_i^{(r)} \cup X_k^{(t)})
    \cup G_{jk}^{(w)}(X_j^{(s)} \cup X_k^{(t)})
    \tag{\dag}
    \label{lec7eq:dag}
  .\]
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/d30a09f107aa40ae.png}
  \end{center}
  Given a $4$-partite $3$-uniform hypergraph $H$ on $X_1 \cup X_2 \cup X_3 \cup
  X_4$, we obtain an induced $3$-partite $3$-uniform hypergraph on $X_i^{(r)} \cup
  X_j^{(s)} \cup X_t^{(t)}$ with edges given by $E(H) \cap
  \triangles\eqref{lec7eq:dag}$.

  The pair $(G, H)$ is a \gls{chain}.
  The \emph{associated chain decomposition} (ACD) of $H$ is the set of all chains
  arising from a given decomposition.
\end{fcdefn}

\begin{fcdefn}[]
  \glsadjdefn{acdquasi}{quasirandom}{}%
  % Definition 2.16
  \label{defn:2.16}
  Let $H$ be a $4$-partite $3$-uniform hypergraph on $X_1 \cup X_2 \cup X_3 \cup
  X_4$, and suppose we have a decomposition of $K(X_1 \cup X_2 \cup X_3 \cup
  X_4)$.
  Then the \gls{acd} of $H(X_i \cup X_j \cup X_k)$ is said to be \emph{$(\eps, \eta,
  \psi)$-quasirandom} if for all but $\eps |X_i||X_j||X_k|$ edges $(x_i, x_j,
  x_k)$, the associated tripartite chain in which $(x_i, x_j, x_k)$lies is
  $(\eta, \psi)$-\gls{chainquasi}.
  The \gls{acd} of $H$ is $(\eps, \eta, \psi)$-quasirandom if all four parts
  (e.g. $H(X_1 \cup X_2 \cup X_3)$) are.
\end{fcdefn}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/25e60913d37f4cb1.png}
\end{center}

\begin{fcthm}[]
  % Theorem 2.17
  \label{thm:2.17}
  Let $H$ be a $4$-partite $3$-uniform hypergraph on $X \cup Y \cup Z \cup W$,
  and let $\eps, \eta > 0$.
  Then there is a decomposition of $K(X \cup Y \cup Z \cup W)$ such that the
  \gls{acd} of $H$ is $(\eps, \eta, \psi)$-\gls{acdquasi}.
  Moreover the number of bipartite graphs in the decomposition is bounded by a
  function of $\eps$ and $\eta$ only, and the number of sets in the partitions
  of $X, Y, Z, W$ depends on $\eps, \eta, \psi$.
\end{fcthm}