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\newpage
\section{Szemerédi Regularity Lemma}

\subsubsection*{Informal statement}

Let $G$ be a graph, $e(G) = \alpha n^2$.
We can partition $V(G)$ into equal parts
\[
  V(G)
  = V_1 \cup \cdots \cup V_k
\]
where $k \le L(\eps)$ depends only on a ``coarseness'' parameter $\eps > 0$, so
that for all but $\eps {k \choose 2}$pairs $1 \le i < j \le k$, the graph
between $V_i$, $V_j$ ``looks'' quasi-random.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/6a0f19fa4d31482f.png}
\end{center}
So, in a sense, ``every graph is somewhat random''.

\begin{fcdefn}[Edge density]
  \glssymboldefn{density}%
  Let $G$ be a graph.
  Let $U, V \subset V(G)$.
  First define
  \[
    d(U, V)
    = \frac{e(U, V)}{|U||V|}
  .\]
\end{fcdefn}

\begin{fcdefn}[$eps$-uniform]
  \glsadjdefn{epsunif}{$\eps$-uniform}{pair}%
  We say that $(U, V)$ is \emph{$\eps$-uniform} if $\forall U_0 \subset U$ and
  $V_0 \subset V$,
  \[
    |\density(U_0, V_0) - d(U, V)|
    \le \eps
  \]
  whenever $|U_0| \ge \eps |U|$ and $|V_0| \ge \eps |V|$.
\end{fcdefn}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/b277b8c449544eeb.png}
\end{center}

\textbf{Fact:} Let $G$ be a graph and let $U, V \subset V(G)$ disjoint which are
\gls{epsunif}.
Let $p = \density(U, V)$.
Then
\begin{align*}
  |\{x \in U : |N(x) \cap V| > (p + \eps) |V|\}|
  &\le \eps|U| \\
  |\{x \in U : |N(x) \cap V| < (p + \eps) |V|\}|
  &\le \eps|U|
\end{align*}

\begin{proof}
  Simply note that if
  \[
    S
    = \{x \in U : |N(x) \cap V| > (p + \eps)|V|\}
  ,\]
  we have $\density(S, V) > (p + \eps)$, hence $|S| < \eps|U|$ due to
  \glsref[epsunif]{uniformity}.

  Second case is the same.
\end{proof}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/a211250b958f49b3.png}
\end{center}

\begin{fclemma}[Counting lemma for triangles]
  Assuming:
  - $G$ a graph on vertex set $V_1 \cup V_2 \cup V_3$ with $V_i$ disjoint
  - $|V_i| = n$
  - $\eps < \frac{p}{2}$
  - $d(V_i, V_j) \ge p$ and $(V_i, V_j)$ is \gls{epsunif}
  Then:
  the number of triangles in $G$ is
  \[
    \ge (1 - 2\eps)(p - \eps)^3 n^3
  .\]
\end{fclemma}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/5c604757cf9948b3.png}
\end{center}

\begin{proof}
  Let
  \[
    V'
    = \{x \in V_1 : |N(x) \cap V_2|, |N(x) \cap V_3| \ge (p - \eps)n\}
  .\]
  Note that $|V'| \ge (1 - 2\eps)n$.
  Now fix $x \in V$ and note
  \begin{align*}
    e(N(x) \cap V_2, N(x) \cap V_3)
    &\ge (p - \eps)|N(x) \cap V_2||N(x) \cap V_3| \\
    &\ge (p - \eps)(p - \eps)(p - \eps)n^2
  \end{align*}
  using the \gls{epsunif} and $p \ge 2\eps$.
  So summing over all $x \in V'$ gives us
  \[
    (1 - 2\eps)(p - \eps)^3 n^3
  \]
  triangles.
\end{proof}

\begin{fcdefn}[Equipartition]
  \glsnoundefn{equip}{equipartition}{equipartitions}%
  Say that $V(G) = V_1 \cup \cdots \cup V_k$ is an \emph{equipartition} if
  $||V_i| - |V_j|| \le 1$ for all $i, j$.
\end{fcdefn}

\begin{fcthm}[Szemerédi Regularity Lemma]
  \label{thm:regularity}
  Assuming:
  - $\eps > 0$, $l \in \Nbb$
  Then:
  there exists $L = L(\eps, l)$ such that the following holds:

  Let $G$ be a graph.
  Then there is an \gls{equip} of $V(G) = V_1 \cup \cdots \cup V_k$, where $l
  \le k \le L(l, \eps)$ and such that all but $\eps{k \choose 2}$ pairs $1 \le i
  < j \le k$, we have that $(V_i, V_j)$ is \gls{epsunif}.
\end{fcthm}

\begin{remark*}
  \leavevmode
  \begin{itemize}
    \item Technically, the \nameref{thm:regularity} also applies when $e(G) =
      o(n^2)$, but often it does not really tell us anything.
    \item The dependence on $\eps$ of $L(\eps)$ is of tower type, i.e.
      \[
        L(\eps)
        \le \ub{2^{2^{2^{\iddots^2}}}}_{\eps^{-C}}
      .\]
      Gowers proved that
      \[
        L(\eps)
        \ge \ub{2^{2^{2^{\iddots^2}}}}_{\eps^{c}}
      .\]
      So these tower bounds are necessary.
  \end{itemize}
\end{remark*}

\begin{fcthm}[The triangle Removal Lemma]
  \label{thm:TRL}
  For all $\eps > 0$, there exists $\delta > 0$ so that the following holds:

  Every graph $G$ on $n$ vertices with $\le \delta n^3$ triangles contains $T
  \subset E(G)$ with $|T| \le \eps n^2$ and $G - T \not\supset K_3$.
\end{fcthm}

\begin{proof}
  Let $\eps > 0$ be given, let $L = L \left( \frac{10}{\eps}, \frac{\eps}{4}
  \right)$ and define $\delta = \frac{\eps^3}{16L^3}$.
  Let $G$ be a graph with $\le \delta n^3$ triangles.
  \begin{center}
    \includegraphics[width=0.6\linewidth]{images/24ee8a352df241b2.png}
  \end{center}
  Apply \nameref{thm:regularity} to $G$ to find $V(G) = V_1 \cup \cdots \cup
  V_k$, with $k \le L \left( \frac{10}{\eps}, \frac{\eps}{4} \right)$.
  Now define
  \begin{align*}
    T
    &= \{\text{all edges between $V_i$, $V_j$ whenever $\density(V_i, V_j) <
    \frac{\eps}{2}$}\} \\
    &\phantom{=} \cup \{\text{all edges between $V_i$, $V_j$ such that $(V_i,
    V_j)$ not \glsref[epsunif]{$\frac{\eps}{4}$-uniform}}\} \\
    &\phantom{=} \cup \{\text{all edges inside of the parts $V_i$}\}
  \end{align*}
  We note
  \begin{align*}
    |T|
    &\le {k \choose 2} \cdot \frac{\eps}{2} \left( \frac{n}{k} \right)^2
    + \frac{\eps}{4} {k \choose 2} \left( \frac{n}{k} \right)^2
    + {\frac{n}{k} \choose 2} k \\
    &\le \left( \frac{\eps}{2} \right) \frac{k^2}{2} \left( \frac{n}{k}
    \right)^2
    + \frac{\eps}{8} n^2
    + \frac{n^2}{2k} \\
    &\le \eps n^2
  \end{align*}
  % We now check that $G - T \not\supset K_3$.
  % Suppose we have a triangle:
  % \begin{center}
  %   \includegraphics[width=0.6\linewidth]{images/6619b04d777b42da.png}
  % \end{center}
  % The triangle must span across the $V_i$ (as we deleted all edges inside them).
  %
  % To be continued...