A couple of fixes to minor-closed classes.

diff --git a/adv.tex b/adv.tex
index dace909479d531a74775b153ed58c0a6a8b172e2..eca96fdd55ad425b02579bb984e8cf4af76aad01 100644 (file)
--- a/adv.tex
+++ b/adv.tex
@@ -79,11 +79,11 @@ edge density $\varrho(\cal C)$ of the class is then defined as the infimum of $\
 Let us consider a~non-trivial minor-closed class~${\cal C} = \Forb({\cal H})$
 and a~graph $X\in{\cal H}$ with the minimum number of vertices.
 Obviously, $\Forb({\cal H}) \subseteq \Forb(X)$, because excluding additional
-minors cannot make the class richer. Also, if we denote the number of vertices
+minors can only reduce the class. Also, if we denote the number of vertices
 of~$X$ by~$k$, we have $X\minorof K_k$ and hence $\Forb(X) \subseteq \Forb(K_k)$.
 When we put these two inclusions together, we get ${\cal C} \subseteq \Forb(K_k)$ and
 so $\varrho({\cal C}) \le \varrho(\Forb(K_k))$. It is therefore sufficient to
-bound the density of classes that exclude a~single complete graph.
+bound the density for classes that exclude a~single complete graph only.
 Moreover, our parameter~$k$ is equal to the well-known Hadwiger number:
 
 \defn\id{hadwiger}%
@@ -106,7 +106,7 @@ Hadwiger number of the class.
 
 \proof
 We already know that it is sufficient to prove the theorem for the case when
-${\cal C}$ excludes on the complete graph~$K_k$.
+${\cal C}$ excludes just the complete graph~$K_k$.
 
 We will prove the contrapositive. If $\varrho({\cal C}) > 2h(k)$, then there is some graph
 $G\in{\cal C}$ such that $\varrho(G) > 2h(k)$. This implies that the average degree
@@ -114,12 +114,12 @@ of~$G$ is greater than~$h(k)$, so by the previous theorem $G$~contains a~subdivi
 of~$K_k$ and hence also~$K_k$ as a~minor.
 \qed
 
-\para
-The Mader's original proof of Theorem \ref{maderthm} yields $h(k) \approx 2^{n^2}$, which is
-very coarse. It was however vastly improved later: Kostochka
-\cite{kostochka:lbh} and independently Thomason \cite{thomason:efc} have proven
-that an~average degree $\Omega(k\sqrt{\log k})$ is sufficient to enforce~$K_k$
-as a~minor and that this is the best what we can get. Their result implies:
+\paran{A~better bound}%
+The Mader's original proof of Theorem \ref{maderthm} yields $h(k) \approx 2^{n^2}$,
+but it is possible to obtain a~closer bound. Kostochka \cite{kostochka:lbh} and
+independently Thomason \cite{thomason:efc} have proven that an~average degree
+$\Omega(k\sqrt{\log k})$ is sufficient to enforce~$K_k$ as a~minor and that
+this is the best what we can get. Their result implies:
 
 \cor
 $\varrho({\cal C}) = \O(k\sqrt{\log k})$ whenever ${\cal C}$ is a~minor-closed