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% 03/02/2026 10AM

Note that this proposition is much easier to prove if we use the much stronger
notion of quasirandomness that was introduced at the beginning of this section.

\begin{notation*}
  \begin{align*}
    \delta_{ZW}
    &= \frac{|G(Z, W)|}{|Z||W|} \\
    \delta_{YZW}
    &= \frac{\text{number of triangles in $G(Y, Z, W)$}}{|Y||Z||W|} \\
    \delta_Y(x, x')
    &= \frac{\text{number of $y \in Y$ such that $x \sim y$, $x' \sim y$}}{|Y|} \\
    \delta_{ZW}(x, x')
    &= \frac{\text{number of edges $zw$ in $G(Z, W)$ such that $x \sim z$, $x \sim
    w$, $x' \sim z$, $x' \sim w$}}{|Z||W|}
  \end{align*}
\end{notation*}

\begin{proof}
  \begin{align*}
    &\!\!\!\!\!\!\!\!
    |\Ebb_{y, z, w} k(y, z, w) \Ebb_x f(x, y, z) g(x, y, w) h(x, z, w)|^8 \\
    &\le (\Ebb_{y, z, w} k(y, z, w)^2)^4
    (\Ebb_{y, z, w} |\Ebb_x f(x, y, z) g(x, y, w) h(x, z, w)|^2)^4 \\
    &\le \delta_{YZW}^4 (\Ebb_{x, x'} \Ebb_{y, z, w} \ub{f_{x, x'}(y, z)}_{f(x,
    y, z)f(x', y, z)} g_{x, x'}(y, w) h_{x, x'}(z, w))^4 \\
    &\le \delta_{YZW}^4 \Ebb_{x, x'} (\Ebb_{y, z, w} f_{x, x'}(y, z) g_{x,
    x'}(y, w) h_{x, x'}(z, w))^4 \\
    &= \delta_{YZW}^4 \Ebb_{x, x'} (\Ebb_{z, w} h_{x, x'}(z, w) \Ebb_y f_{x,
    x'}(y, z) g_{x, x'}(y, w))^4
  \end{align*}
  So far, we haven't done anything difficult, but the first tricky step comes
  now.
  It will be important that we do not (immediately) separate $h_{x, x'}(z, w)$
  from the rest of the expression, because it will be important that we use the
  fact that $h_{x, x'}(z, w)$ is only supported on triangles of $G$.
  \begin{align*}
    &\!\!\!\!\!\!\!\!
    \delta_{YZW}^4 \Ebb_{x, x'} (\Ebb_{z, w} h_{x, x'}(z, w) \Ebb_y f_{x,
    x'}(y, z) g_{x, x'}(y, w))^4 \\
    &= \delta_{YZW}^4 \Ebb_{x, x'} (\Ebb_{z, w} h_{x, x'}(z, w) \indicator{G}(z,
    w) \Ebb_y f_{x, x'}(y, z) g_{x, x'}(y, w))^4 \\
    &\le \delta_{YZW}^4 \Ebb_{x, x'} (\Ebb_{z, w} h_{x, x'}(z, w)^2)^2
    (\Ebb_{z, w} \indicator{G}(z, w) (\Ebb_y f_{x, x'}(y, z) g_{x, x'}(y,
    w))^2)^2 \\
    &\le \delta_{YZW}^4 \Ebb_{x, x'} \delta_{ZW}(x, x')^2
    (\Ebb_{y, y'} \Ebb_{z, w} \ub{f_{x, x', y, y'}(z)}_{\defeq f_{x, x'}(y, z)
    f_{x, x'}(y', z)} g_{x, x', y, y'}(w) \indicator{G}(z, w))^2 \\
    &\le \delta_{YZW}^4 \Ebb_{x, x'} \delta_{ZW}(x, x')^2 (\Ebb_{y, y'} \Ebb_w
    g_{x, x', y, y'}(w) \Ebb_z f_{x, x', y, y'}(z) \indicator{G}(z, w))^2 \\
    &= \delta_{YZW}^4 \Ebb_{x, x'} \delta_{ZW}(x, x')^2 (\ub{\Ebb_{y, y'} \Ebb_w
    g_{x, x', y, y'}(w) \indicator{G}(x, q) \indicator{G}(x', y)
    \indicator{G}(x, y') \indicator{G}(x', y') \Ebb_z f_{x, x', y, y'}(z)
    \indicator{G}(z, w)}_{
      = \Ebb_{y, y'} \prod \indicator{G}(x, y) \Ebb_{z, w} f_{x, x', y, y'}(z)
      g_{x, x', y, y'}(w) \indicator{G}(z, w)
    })^2 \\
    &\le \delta_{YZW}^4 \Ebb_{x, x'} \delta_{ZW}(x, x')^2 (\Ebb_{y, y'}
    \prod \indicator{G}) (\Ebb_{y, y'} |\Ebb_{z, w} f_{x, x', y, y'}(z) g_{x,
    x', y, y'}(w) \indicator{G}(z, w)|^2) \\
    &= \delta_{YZW}^4 \Ebb_{x, x'} \delta_{ZW}(x, x')^2 \delta_Y(x, x')^2
    \Ebb_{y, y'} (\Ebb_w g_{x, x', y, y'}(w) \indicator{G}(w, x)
    \indicator{G}(w, x') \indicator{G}(w, y) \indicator{G}(w, y')
    \Ebb_z f_{x, x', y, y'}(z) \indicator{G}(z, w))^2 \\
    &\le \delta_{YZW}^4 \Ebb_{x, x'} \delta_{ZW}(x, x')^2 \delta_Y(x, x')^2
    \Ebb_{y, y'} (\ub{\Ebb_w \indicator{ug}(w, x) \indicator{G}(w, x')
    \indicator{G}(w, y) \indicator{G}(w, y')}_{\eqdef \delta_W(x, x', y, y')})
    \Ebb_w |\Ebb_z f_{x, x', y', y'}(z) \indicator{G}(z, w)|^2 \\
    &= \Ebb_{x, x', y, y', z, z'} \ub{f_{x, x', y, y', z, z'}}_{f_{x, x', y,
    y'}(z) f_{x, x', y, y'}(z')} \phi(x, x', y, y', z, z') \\
    &= \Ebb_{x, x', y, y', z, z'} F(x, x', y, y', z, z' \phi(x, x', y, y', z,
    z') \\
    &= \langle F, \phi \rangle
  \end{align*}
  where
  \begin{align*}
    \phi(x, x', y, y', z, z')
    &= \delta_{YZW}^4 \delta_{ZW}(x, x')^2 \delta_Y(x, x')^2 \delta_W(x, x', y,
    y') \delta_W(x, x', y, y', z, z') \\
    F(x, x', y, y', z, z')
    &= f_{x, x', y, y', z, z'}
  \end{align*}
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/27b418d5b8c84cdd.png}
  \end{center}
  Note that $\phi$ only depends on the underlying graph, which we assumed was
  \gls{epsquasi}.
  We can compute
  \[
    \Ebb_{x, x', y, y', z, z'} \phi(x, x', y, y', z, z')
    = (\delta_{YZ} \delta_{YZ} \delta_{ZW})^4 \cdots
    = (\delta_{XY} \delta_{YZ} \delta_{XZ})^4
    (\delta_{XW} \delta_{YW} \delta_{ZW})^8
    \eqdef \theta
  .\]