Update comparable-box-dimension.tex

comparable-box-dimension.tex

@@ -436,7 +436,7 @@ behave well with respect to full clique-sums.
   sum extendable representation $h$ by comparable boxes in
   $\mathbb{R}^{\max(d_1,d_2)}$.
 \end{lemma}
-The proof is in the appendix, but the idea is to translate (allowing
+The proof is omitted, but the idea is to translate (allowing
 also exchanges of dimensions) and scale $h_2$ to fit in
 $h_1^\varepsilon(C_1)$.
 The following lemma enables us to pick the root clique at the expense of increasing
@@ -446,7 +446,7 @@ the dimension by $\omega(G)$.
   $C^\star$-clique-sum extendable touching representation by comparable
   boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus V(C^\star))$.
 \end{lemma}
-  The proof is also in the appendix, but it essentially the same as the one of
+  The proof is also omitted, but it is essentially the same as the one of
   Lemma~\ref{lemma-apex}.
 The following lemma provides an upper bound on $\ecbdim(G)$ in terms
 of $\cbdim(G)$ and $\chi(G)$.
@@ -618,7 +618,7 @@ $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for suffici
 \end{proof}
 The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree.
 Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree}
-extends to graphs of treewidth at most $k$ (see the proof in the appendix).
+extends to graphs of treewidth at most $k$ (proof omitted).
 \begin{corollary}\label{cor-tw}
 Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$.
 \end{corollary}