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What is the number of triangle-free graphs on $n$ vertices?

\begin{fcthm}[Erdős, Klietman, Rothchild, 70s]
  \label{thm:EKR}
  The number of triangle-free graphs on $n$ vertices is $2^{\frac{n^2}{4} +
  o(n^2)}$.
\end{fcthm}

\begin{remark*}
  ``$\ge$'' comes from considering all subgraphs of $K_{\frac{n}{2},
  \frac{n}{2}}$.
  Can sort of think of this as saying `almost all triangle-free graphs are
  bipartite'.
\end{remark*}

\begin{fclemma}[]
  \label{lemma:TFcontainers}
  For every $\eps > 0$ and $n$ large enough, there exists a family of graphs
  $\mathcal{G} = \mathcal{G}_{n, \eps}$ on $[n]$ with the following properties:
  \begin{enumerate}[(1)]
    \item $|\mathcal{G}| \le 2^{\eps n^2}$.
    \item All $G \in \mathcal{G}$ can be made triangle-free by removing $\le
      \eps n^2$ edges.
    \item Every triangle-free graph on $[n]$ is contained in some $G \in
      \mathcal{G}$.
  \end{enumerate}
\end{fclemma}

\begin{proof}[Proof of \nameref{thm:EKR} using \cref{lemma:TFcontainers}]
  Let $\eps > 0$ be given.
  Then for large enough $n$,
  \begin{align*}
    \text{\# of triangle-free graphs}
    &\le \sum_{G \in \mathcal{C}_{n, \eps^2}} \sum_{H \subset G} 1 \\
    &\le \sum_{G \in \mathcal{C}} 2^{\frac{n^2}{4}} \cdot {{n \choose 2} \choose
    \eps^2 n^2} \\
    &\le 2^{\frac{n^2}{4} + \eps n^2}
    \qedhere
  \end{align*}
\end{proof}

\begin{proof}[Proof of \cref{lemma:TFcontainers}]
  Given a graph $H$ on $[k]$ and a partition $[n] = V_1 \cup \cdots \cup V_k$,
  we define $G$ to be the blow up of $H$ onte $\{V_i\}$ by $V(G) = [n]$ and
  include all of $V_i V_j$ into $G$ whenever $ij \in H$.

  Let $\eps > 0$ be given.
  We define $\mathcal{G} = \mathcal{G}_{n, \eps}$ to be all graphs formed in the
  following steps.
  Let $L = L \left( \frac{10}{\eps}, \frac{\eps^2}{2} \right)$.
  \begin{enumerate}[(1)]
    \item Let $H$ be a triangle-free graph on $[k]$ where $k \le l$.
    \item Let $u[n] = V_1 \cup \cdots \cup V_k$ be an \gls{equip}, and blow up
      $H$ onto $\{V_i\}$.
    \item Throw in $\eps^2 n^2$ edges in any way.
  \end{enumerate}
  Check $|\mathcal{G}|$ is small:
  \[
    |\mathcal{G}|
    \le \ub{2^{L^2}}_{\text{step 1}} \ub{(n! 2^k)}_{\text{step 2}} \ub{{n^2
    \choose \eps^2 n^2}}_{\text{step 3}}
    \le 2^{\eps n^2}
  .\]
  Each $G \in \mathcal{G}$ can be made triangle-free by removing $\le \eps n^2$
  edges: this is true by construction (in fact only need to remove $\eps^2 n^2$
  edges).

  Finally, we need to check that every triangle-free graph on $[n]$ is contained
  in some $G \in \mathcal{G}$.
  We apply \nameref{thm:regularity} with parameter $l = \frac{10}{\eps}$,
  $\frac{\eps^2}{2}$, to get a partition $[n] = V_1 \cup \cdots \cup V_k$ with
  $k \le L$.
  Consider the reduced graph $R = R_{\eps^2, \frac{\eps^2}{2}}$.
  Note $R$ is triangle-free.
  So use $G \subset R$ blown up to $\{V_1\}$ plus $\eps^2 n$ extra edges.
\end{proof}

Reminder of \nameref{thm:regularity}:

\begin{fcthm}[Regularity Lemma]
  For $\eps > 0$, $l \in \Nbb$, there exists $L(l, \eps)$ such that the
  following holds:

  Let $G$ be a graph.
  Then there exists an \gls{equip} $V(G) = V_1 \cup \cdots \cup V_k$ with $l \le
  k \le L(l, \eps)$, where $\{V_i\}$ \gls{equip} and all but $\eps {k \choose
  2}$ pairs $(V_i, V_j)$ are \gls{epsunif}.
\end{fcthm}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/108b5b70772e446d.png}
\end{center}

\begin{proof}[Proof of \nameref{thm:regularity}]
  Given $\eps, l$ and a graph $G$, we loop the following, iteratively refining a
  partition.

  Start with $V(G) = V_1 \cup \cdots \cup V_l$, arbitrarily into an \gls{equip}.
  At a general step, we are given
  \[
    V(G)
    = V_1 \cup \cdots \cup V_k
  \]
  an \gls{equip}.

  If there are $\le \eps {k \choose 2}$ non \gls{epsunif} pairs, then output
  $\{V_i\}_{i = 1}^k$ and done
  So assume $> {k \choose 2}$.
  For each non \gls{epsunif} pair $V_i, V_j$, we select $W_{ij} \subset V_i$,
  $W_{ji} \subset V_j$, $W_{ij}|, |W_{ji}| \ge \eps|V_i|$ and $|\density(W_{ij},
  W_{ji}) - d(V_i, V_j)| > \eps$.

  Now for each $V_i$ consider $\{W_{ij}\}_{i = 1}^k$.
  Let
  \[
    V_i
    = \bigcup_j U_{ij}
  \]
  be a common refinement of these $\{W_{ij}\}$.
  We now refine this partition further to an \gls{equip} where each cell has
  size $\frac{n}{k4^k}$ (this is a slight cheat $*$).
  Putting all these partitions together, we get
  \[
    V(G)
    = \bigcup_{i, j} V_{ij}
  .\]
  Now loop again with this partition.

  Next step: show that this loop eventually terminates.
