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\title{Drafts of Proofs for 4.2 and 4.3}

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\section{Theorem 4.2}

Some notes:
\begin{enumerate}
    \item There are more details to the proof in the appendix.
    \item The theorem really hinges on Lemma 4.2, the proof of A.2 is just using properties of matrix norms.
    \item We may need to prove/find a source for this cut norm being a norm (would simplify the proof of A.2 a bit).
\end{enumerate}

\subsection{Lemma A.2}



\subsection{Cut norm is a norm.}



\subsection{Theorem 4.2 proof.}

\section{Theorem 4.3}

Some notes:
\begin{enumerate}
    \item The theorem is really Lemma A.3 with certain choices for $F$, $W$.
    \item Lemma A.3 is Theorem 2.5 in \cite{lov-sze} in this paper: \href{https://arxiv.org/pdf/math/0408173.pdf}{ArXiv link} \cite{lov-sze}, the proof starts on page 16.
    \item The coefficient in A.3 should be 1/18, not 1/8 (as far as we can tell).
    % \item Lemma A.3 should follow from Azuma's inequality (reference?), the main content in the proof is setting up the hypotheses of the inequality.
\end{enumerate}

% \subsection{Lemma A.3}

% From \cite{lov-sze}, the key steps are:
% \begin{itemize}
%     \item We want to apply Azuma's inequality: \href{https://en.wikipedia.org/wiki/Azuma's_inequality}{Azuma's Inequality Wikipedia}. We need:
%     \begin{itemize}
%         \item A \textbf{martingale} $X_m$, $m \in \mathbb{N}$. See \href{https://en.wikipedia.org/wiki/Martingale_(probability_theory)}{Wikipedia on Martingales}.
%         \item A bound on the differences between consecutive martingales: $| X_m - X_{m-1} | \leq c_m$.
%     \end{itemize}

%     \item Construct a sequence of graphs $G_1, \ldots, G_m, \ldots$. At each step we add one new node and edges between the $v(G_{m-1})$ and the new node.
%     \item Th
% \end{itemize}


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