diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex
index 94bf6043b8a866fa712b50ba46292161524e5c89..a4e7bf6c160b11f4f3c5971d1d1e7812940be9eb 100644
--- a/comparable-box-dimension.tex
+++ b/comparable-box-dimension.tex
@@ -5,6 +5,7 @@
 \usepackage{amssymb}
 \usepackage{colonequals}
 \usepackage{epsfig}
+\usepackage{tikz}
 \usepackage{url}
 \newcommand{\GG}{{\cal G}}
 \newcommand{\HH}{{\cal H}}
@@ -23,6 +24,7 @@
 \newcommand{\vol}{\brm{vol}}
 %%%%%
 
+\newcommand{\note}[1]{\textcolor{blue}{#1}}
 
 \newtheorem{theorem}{Theorem}
 \newtheorem{corollary}[theorem]{Corollary}
@@ -37,7 +39,7 @@
 Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.}\and
 Daniel Gon\c{c}alves\thanks{...}\and 
 Abhiruk Lahiri\thanks{...}\and
-Jane Tan\thanks{...}\and
+Jane Tan\thanks{Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK. E-mail: {\tt jane.tan@maths.ox.ac.uk}}\and
 Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology.  E-mail: {\tt torsten.ueckerdt@kit.edu}}}
 \date{}
 
@@ -45,9 +47,7 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology.  E-mail: {\tt torsten
 \maketitle
 
 \begin{abstract}
-The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented
-as a touching graph of comparable boxes in $\mathbb{R}^d$ (two boxes are comparable if one of them is
-a subset of a translation of the other one).  We show that proper minor-closed classes have bounded
+Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other. The comparable box dimension of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded
 comparable box dimension and explore further properties of this notion.
 \end{abstract}
 
@@ -58,18 +58,17 @@ if there exists a function $f:V(G)\to \OO$ (called a \emph{touching representati
 such that for distinct $u,v\in V(G)$, the interiors of $f(u)$ and $f(v)$ are disjoint
 and $f(u)\cap f(v)\neq\emptyset$ if and only if $uv\in E(G)$.
 Famously, Koebe~\cite{koebe} proved that a graph is planar if and only if it is a touching graph of balls in $\mathbb{R}^2$.
-This result motivated a number of strengthenings and variations~\cite{...}; most relevantly for us, every planar graph is
-a touching graph of cubes in $\mathbb{R}^3$~\cite{felsner2011contact}.
+This result motivated a number of strengthenings and variations (see \cite{graphsandgeom, sachs94} for some classical examples); most relevantly for us, every planar graph is a touching graph of cubes in $\mathbb{R}^3$~\cite{felsner2011contact}.
 
-An attractive feature of touching representation is that it makes it possible to represent graph classes that are sparse
-(e.g., planar graphs, or more generally, graph classes with bounded expansion theory~\cite{nesbook}),
-whereas in a general intersection representation, the represented class always includes arbitrarily large cliques.
+An attractive feature of touching representations is that it is possible to represent graph classes that are sparse
+(e.g., planar graphs, or more generally, graph classes with bounded expansion theory~\cite{nesbook}).
+This is in contrast to general intersection representations where the represented class always includes arbitrarily large cliques.
 Of course, whether the class of touching graph of objects from $\OO$ is sparse or not depends on the system $\OO$.
 For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis-aligned boxes in $\mathbb{R}^3$, where the vertices in
 one part are represented by $m\times 1\times 1$ boxes and the vertices of the other part are represented by $1\times n\times 1$
-boxes (a \emph{box} is the Cartesian product of intervals of non-zero length).
+boxes (a \emph{box} is the Cartesian product of intervals of non-zero length, in particular axis-aligned).
 Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} noticed that this issue disappears if we forbid such a combination of
-long and wide boxes: For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if a translation of $B_1$ is a subset of $B_2$.
+long and wide boxes: For two boxes $B_1$ and $B_2$, we write $B_1 \sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$.
 We say that $B_1$ and $B_2$ are \emph{comparable} if $B_1\sqsubseteq B_2$ or $B_2\sqsubseteq B_1$.
 A \emph{touching representation by comparable boxes} of a graph $G$ is a touching representation $f$ by boxes
 such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable.  For a graph $G$, let the \emph{comparable box dimension} $\cbdim(G)$
@@ -102,12 +101,13 @@ or expressible in the first-order logic~\cite{logapx}.
 \section{Operations}
 
 Let us start with a simple lemma saying that the addition of a vertex increases the comparable box dimension by at most one.
-In particular, this implies that $\cbdim(G)\le |V(G)$.
+In particular, this implies that $\cbdim(G)\le |V(G)|$.
 \begin{lemma}\label{lemma-apex}
 For any graph $G$ and $v\in V(G)$, we have $\cbdim(G)\le \cbdim(G-v)+1$.
 \end{lemma}
 \begin{proof}
 Let $f$ be a touching representation of $G-v$ by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G-v)$.
+We define a representation $h$ of $G$ as follows.
 For each $u\in V(G)\setminus\{v\}$, let $h(u)=[0,1]\times f(u)$ if $uv\in E(G)$ and 
 $h(u)=[1/2,3/2]\times f(u)$ if $uv\not\in E(G)$.  Let $h(v)=[-1,0]\times [-M,M] \times \cdots \times [-M,M]$,
 where $M$ is chosen large enough so that $f(u)\subseteq [-M,M] \times \cdots \times [-M,M]$ for every $u\in V(G)\setminus\{v\}$.
@@ -120,7 +120,7 @@ For a box $B=I_1\times \cdots \times I_d$ and $i\in\{1,\ldots,d\}$, let $B[i]=I_
 If $G$ has a touching representation $f$ by comparable boxes in $\mathbb{R}^d$, then $\omega(G)\le 2^d$.
 \end{lemma}
 \begin{proof}
- For any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty intersection.
+To represent any clique $A = \{a_1,\ldots,a_w\}$ in $G$, the corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty intersection.
  Since axis-aligned boxes have the Helly property, there is a point $p \in \mathbb{R}^d$ contained in $f(a_1) \cap \cdots \cap f(a_w)$.
  As each box is full-dimensional, its interior intersects at least one of the $2^d$ orthants at $p$.
  Since $f$ is a touching representation, $f(a_1),\ldots,f(a_d)$ have pairwise disjoint interiors and hence $w \leq 2^d$.
@@ -138,11 +138,11 @@ For each vertex $v\in V(G)$, let $p(v)$ be the node $x\in V(T)$ such that $v\in
 The \emph{adhesion} of the tree decomposition is the maximum of $|\beta(x)\cap\beta(y)|$ over distinct $x,y\in V(T)$,
 and its \emph{width} is the maximum of the sizes of the bags minus $1$.  The \emph{treewidth} of a graph is the minimum
 of the widths of its tree decompositions.  We will need to know that graphs of bounded treewidth have bounded comparable box dimension.
-In fact, we will prove the following stronger fact (TODO: Was this published somewhere before? I am only aware of the upper bound of $t+2$ on the boxicity of $G$~\cite{box-treewidth}.)
+In fact, we will prove the following stronger fact. \note{(TODO: Was this published somewhere before? I am only aware of the upper bound of $t+2$ on the boxicity of $G$~\cite{box-treewidth}.)}
 
 \begin{lemma}\label{lemma-tw}
 Let $(T,\beta)$ be a tree decomposition of a graph $G$ of width $t$.
-Then $G$ has a touching representation $h$ by axis-aligned hypercubes in $R^{t+1}$ such that
+Then $G$ has a touching representation $h$ by axis-aligned hypercubes in $\mathbb{R}^{t+1}$ such that
 for $u,v\in V(G)$, if $p(u)\prec p(v)$, then $h(u)\sqsubset h(v)$.
 Moreover, the representation can be chosen so that no two hypercubes have the same size.
 \end{lemma}
@@ -153,7 +153,7 @@ and that for each $x\in V(T)$ with parent $y$, we have $|\beta(x)\setminus\beta(
 we can subdivide the edge $xy$, introducing $|\beta(x)\setminus\beta(y)|-1$ new vertices,
 and set their bags appropriately).  It is now natural to relabel the vertices of $G$
 so that $V(G)=V(T)$, by giving the unique vertex in $\beta(x)\setminus\beta(y)$
-the label $x$.  In particular, $p(x)=x$.  Furthermore, we can assume that $y\in\beta(x)$.
+the label $x$.  In particular, $p(x)=x$.  Furthermore, switching to this new labeling, we can assume that $y\in\beta(x)$.
 Otherwise, if $y$ is not the root of $T$, we can replace the edge $xy$ by the edge from $x$
 to the parent of $y$.  If $y$ is the root of $T$, then the subtree rooted in $x$ induces a
 union of connected components in $G$, and we can process this subtree separately from the
@@ -211,7 +211,7 @@ Hence, $\min(h(w)[j])-2\epsilon\le \min(h''(w_1)[j])\le \max(h''(w_1)[j])\le \ma
 Since $h(x_i)[j]\subseteq h''(w_1)[j]$ by (a) and the length of the interval $h(x_i)[j]$ is greater than $2\varepsilon$,
 we conclude that $h(x_i)[j]\cap h(w)[j]\neq\emptyset$.  Therefore, the boxes $h(w)$ and $h(x_i)$ touch.
 
-Consider now two non-adjacent vertices of $G$, say $x_i$ and $w$.  As we noted before, if $x_i$ and $w$ are
+Finally, consider two non-adjacent vertices of $G$, say $x_i$ and $w$.  As we noted before, if $x_i$ and $w$ are
 incomparable in $\prec$, then (a) and (b) implies that the boxes $h(x_i)$ and $h(w)$ are disjoint.
 Suppose now that say $x_i\prec w$, and let $j=\varphi(w)$.  If $w\in\beta(x_i)$, then $h(x_i)[j]$ and $h(w)[j]$ are disjoint by (i).
 Otherwise, let $y$ be the last vertex on the path from $w$ to $x_i$ in $T$ such that $w\in\beta(y)$ and let $z$ be the child of $y$
@@ -310,7 +310,7 @@ $xy\in E(T_\beta)$.  Hence, again we have $f(u)\cap f(v)\neq\emptyset$.
 We can now combine Theorem~\ref{thm-cs} with Lemma~\ref{lemma-cliq}.
 \begin{corollary}\label{cor-cs}
 If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then there exists a graph $G'$ such that
-$G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}$.
+$G\subseteq G'$ and \[\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}.\]
 \end{corollary}
 
 Note that only bound the comparable box dimension of a supergraph
@@ -407,7 +407,7 @@ Any graph $G$ is a subgraph of the strong product of a path, a graph of threewid
 \item $t=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$.
 \end{itemize}
 Moreover, for every $k$, there exists $t$ such that if $K_k\not\preceq_m G$, then $G$ is a subgraph of a clique-sum
-for graphs that can be obtained from the strong product of a path and a graph of treewidth at most $t$ by adding
+of graphs that can be obtained from the strong product of a path and a graph of treewidth at most $t$ by adding
 at most $t$ apex vertices.
 \end{theorem}
 
@@ -420,13 +420,10 @@ comparable box dimension at most $t+2+\lceil \log_2 m\rceil$.
 Without loss of generality, we can assume that $k=\log_2 m$ is an integer and the
 vertices of $K_m$ are elements of $\{0,1\}^k$.  Moreover, we can assume that $V(P)=\{1,\ldots,n\}$
 with $ij\in E(P)$ iff $|i-j|=1$.  Let $h$ be the touching representation of $T$ by
-hypercubes in $\mathbb{R}^{t+1}$ obtained by Lemma~\ref{lemma-tw}.
-
-The representation $f$ of $P\boxtimes T\boxtimes K_m$ in $\mathbb{R}^{t+2+k}$
-is obtained as follows: For $p\in V(P)$, $v\in V(T)$, and $(x_1,\ldots, x_k)\in V(K_m)$,
+hypercubes in $\mathbb{R}^{t+1}$ obtained by Lemma~\ref{lemma-tw}. Then an explicit representation $f$ of $P\boxtimes T\boxtimes K_m$ in $\mathbb{R}^{t+2+k}$ can be obtained as follows. For $p\in V(P)$, $v\in V(T)$, and $(x_1,\ldots, x_k)\in V(K_m)$,
 we set
 $$f(p,v,x_1, \ldots, x_k)[j]=\begin{cases}
-f(v)[j]&\text{ for $j=1,\ldots, t+1$}\\
+h(v)[j]&\text{ for $j=1,\ldots, t+1$}\\
 [p,p+1]&\text{ if $j=t+2$}\\
 [x_{j-t-2},x_{j-t-2}+1]&\text{ if $j>t+2$.}
 \end{cases}$$
@@ -444,7 +441,7 @@ Moreover, for every $k$ there exists $d$ such that if $K_k\not\preceq_m(G)$, the
 $\cbdim(G)\le d$.
 \end{corollary}
 Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
-and comparable box dimension $n$, showing that the dependence of the comparable box dimension cannot be
+and comparable box dimension $n$. It follows that the dependence of the comparable box dimension cannot be
 subpolynomial (though the degree $\log_2 81$ of the polynomial established in Corollary~\ref{cor-minor}
 certainly can be improved).  The dependence of the comparable box dimension on the size of the forbidden minor that we
 established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}.
diff --git a/data.bib b/data.bib
index 3ce675d0e15f0a5ecf9bf5b2492ce12098556234..dfa99649d4e19ae478aa06414bbbf7c9ec954793 100644
--- a/data.bib
+++ b/data.bib
@@ -5254,7 +5254,7 @@ note="In Press"
 @article{mohar2002coloring,
   title={Coloring Eulerian triangulations of the projective plane},
   author={Mohar, Bojan},
-  journal={Discrete mathematics},
+  journal={Discrete Mathematics},
   volume=244,
   pages={339--344},
   year=2002
@@ -5334,3 +5334,19 @@ note = {In Press}
   doi       = {https://doi.org/10.1016/j.jctb.2006.12.004},
   url       = {https://www.sciencedirect.com/science/article/pii/S0095895607000111},
 }
+
+@book{graphsandgeom,
+  author= {L. Lov\'asz},
+  title={Graphs and Geometry},
+  publisher={American Mathematical Society},
+  year=2019,
+  address   = "Providence"}
+
+@article{sachs94,
+  author    = {Horst Sachs},
+  title     = {Coin graphs, polyhedra, and conformal mapping },
+  journal   = {Discrete Mathematics},
+  volume    = {134},
+  year=1994,
+  pages={133--138}}
+}