%! TEX root = PC.tex
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% 03/11/2025 09AM
%
% \begin{fcthm}[]
%   There exists a constant $C > 0$ such that
%   \[
%     R(H)
%     \le C2^d k
%   ,\]
%   for all bipartite graphs $H$ on $k$ vertices with max degree $d$.
% \end{fcthm}

\begin{proof}[Proof of \nameref{thm:foxsudakovconlon}]
  Apply our second dependent random choice lemma
  (\cref{lemma:secondrandomchoice}) to find an \approxskrich{d}{k} set $U
  \subset V(G)$, with $|U| > 2k$, where $G$ is the graph of the denser colour.
  Then apply the embedding lemma for \approxskrich{d}{k} sets
  (\cref{lemma:approxembed}) to find $G \supset H$.
\end{proof}

\begin{fcthm}[Graham, Rödl, Ruciński]
  \label{thm:grr}
  Assuming:
  - $H$ a graph on $k$ vertices with max degree $d$
  Then:
  there exists a constant $C_d > 0$ depending only on $d$, such that
  \[
    R(H)
    \le C_d \cdot k
  .\]
  We will show
  \[
    C_d
    \le 2^{Cd(\log d)^2}
  \]
  for some absolute constant $C$.
\end{fcthm}

\begin{remark*}
  There exists $H$ such that $R(H) > 2^{cd} \cdot k$, for $c > 0$.

  This is also ``almost'' the best bound known on these Ramsey numbers.
  The best known bound is
  \[
    R(H)
    \le 2^{Cd(\log d)} \cdot k
  ,\]
  which was proved by Fox, Conlon, Sudakov.
\end{remark*}

\begin{fcdefn}[(p,eps)-dense]
  \glsadjdefn{pepsdense}{$(p, \eps)$-dense}{graph}%
  Say that a graph $G$ on $n$ vertices is \emph{$(p, \eps)$-dense} if $\forall
  A, B \subset V(G)$, $|A|, |B| \ge \eps n$, we have
  \[
    e(A, B)
    \ge p|A||B|
  .\]
\end{fcdefn}

\begin{fclemma}[Embedding lemma for $(p, eps)$-dense graphs]
  \label{lemma:grrembedding}
  Assuming:
  - $\eps = \frac{p^d}{2d}$
  - $n \ge \frac{k}{\eps}$
  - $G$ a graph on $n$ vertices which is \gls{pepsdense}
  - $H$ a graph on $k$ vertices with maximum degree $\le d$
  Then:
  $G \supset H$.
\end{fclemma}

\textbf{Sketch of how to use embedding lemma:}
Apply lots of times, with say $p = \frac{1}{4d}$.
Get lots of parts, with high density between all.
Then can just greedily embed.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/11327bfed74d4b64.png}
\end{center}

The lemma for greedily embedding will be:

\begin{fclemma}[]
  \label{lemma:grrgreedy}
  Assuming:
  - $G$ a graph on $n > 2k$ vertices
  - $\delta(G) \ge \left( 1 - \frac{1}{4d} \right) n$
  - $H$ a graph on $k$ vertices with max degree $\le d$
  Then:
  $G \supset H$.
\end{fclemma}

With the statements of these lemmas, we can already get an idea of where the
$2^{C d(\log d)^2}$ comes from.
We will use $p = \frac{1}{4d}$, so by \nameref{lemma:grrembedding}, we can
always either find a copy of $H$, or zoom in by a factor of $2^{C' d \log d}$ to
get a very dense bipartite graph.
We will apply this $C''\log d$ times to get a highly dense graph, which we can then
apply \cref{lemma:grrgreedy} to, and so we will need at least $(2^{C' d \log
d})^{C'' \log d} = 2^{C d (\log d)^2}$ vertices.

We now focus on discussing and proving \nameref{lemma:grrembedding}.

\textbf{Observation:}
Let $G$ be a \gls{pepsdense} graph, and let $C_0, C_1, \ldots, C_d \subset V(G)$
with $|C_0| \ge d\eps n$ and $|C_i| > \eps n$.
Then there exists $x \in C_0$ such that
\[
  |N(x) \cap C_i|
  \ge p|C_i|
\]
for all $i = 1, \ldots, d$.

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/ffcb0e1fca63491a.png}
\end{center}

\begin{proof}
  Let $B_i = \{x \in C_0 : |N(x) \cap C_i| < p|C_i|\}$.
  Note
  \[
    e(B_i, C_i)
    < p|B_i||C_i|
  .\]
  Hence $B_i < \eps n$ for all $i$.
  So we must have
  \[
    C_0 \setminus (B_1 \cup \cdots \cup B_d)
    \neq \emptyset
    .
    \qedhere
  \]
\end{proof}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/180fd11c30764b26.png}
\end{center}

\begin{proof}[Proof of \nameref{lemma:grrembedding}]
  The idea of the proof is to embed the vertices one by one.
  This could go wrong if we place vertices in a way that reduces the
  possibilities for future vertices by too much.
  We avoid this issue by embedding each vertex using the above Observation, to
  make sure that future vertices still have lots of valid places to be embedded
  into.

  Let $V(H) = \{x_1, \ldots, x_k\}$.
  We inductively map $x_i \to v_i$, $v_i \in V(G)$.
  At each stage we maintain a set of candidates for each $x_i$.
  In particular,
  \[
    C_i(j) = \{\text{candidates for $x_j$ at step $i$}\}
  ,\]
  meaning after we have embedded $x_1 \to v_1$, \ldots, $x_i \to v_i$, we
  maintain that after embedding $x_1, \ldots, x_i$ we have
  \[
    C_i(j)
    \subset \bigcap_{\substack{r \le i \\ x_j \sim x_r}} N(v_r)
    \tag{$*$}
    \label{eq:lec11subsetcond}
  .\]
  We also require
  \[
    |C_i(j)|
    \ge p^{d_i(j)} n
    \tag{$**$}
    \label{eq:lec11sizecond}
  \]
  where $d_i(j) = |N(x_j) \cap \{x_1, \ldots, x_i\}|$.

  Say at step $i$, I have mapped $x_1, \ldots, x_i$ preserving adjacencies and
  preserving \eqref{eq:lec11subsetcond} and \eqref{eq:lec11sizecond}.

  We now choose $v_{i + 1}$.
  We want to choose $v_{i + 1} \in C_i(i + 1)$.
  Note
  \[
    |C_i(i + 1)|
    \ge p^d n
    \ge 2d\eps n
  .\]
  Let $C_0 = C_i(i + 1) \setminus \{v_1, \ldots, v_i\}$ and note
  \[
    |C_0|
    \ge 2d \eps n - k
    \ge (2d - 1) \eps n
  .\]
  Now apply the observation with $C_0$ and $C_i(j_1), \ldots, C_i(j_d)$, where
  $x_i \sim x_{j_1}, \ldots, x_{j_d}$, to find a vertex $v_{i + 1} \in C_i(i +
  1)$ such that
  \begin{align*}
    |N(v_{i + 1}) \cap C_i(j_1)|
    &\ge p \cdot p^{d_i(j_1)} \cdot n \\
    |N(v_{i + 1}) \cap C_i(j_2)|
    &\ge p \cdot p^{d_i(j_2)} \cdot n \\
    &~\vdots \\
    |N(v_{i + 1}) \cap C_i(j_d)|
    &\ge p \cdot p^{d_i(j_d)} \cdot n
  \end{align*}
  as desired.
\end{proof}