From acbf6fc1dbd122d5788ac018c90dc2d928a8a3b3 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 24 Aug 2022 22:33:02 +0200
Subject: More examples

---
 sections/examples.tex | 74 ++++++++++++++++++++++++++++++++-------------------
 1 file changed, 46 insertions(+), 28 deletions(-)

(limited to 'sections/examples.tex')

diff --git a/sections/examples.tex b/sections/examples.tex
index f65ebce..5f75953 100644
--- a/sections/examples.tex
+++ b/sections/examples.tex
@@ -10,11 +10,12 @@
 	\end{enumerate}
   \end{example}
 
-  $\cL$ of course has HP and is essentialy countable. JEP is also easy,
-  as having two finite linear orderings we can just embed the one with less
+  $\cL$ of course has HP and is essentially countable. JEP is also easy,
+  as having two finite linear orderings we can just embed the one with fewer 
   elements into the bigger one.
 
-  We will show that $\cL$ has CAP. Let $C$ be a finite linear ordered set.
+  We will show that $\cL$ has canonical amalgamation (CAP). 
+  Let $C$ be a finite linear ordered set.
   We will define $\otimes_C$. Let $A$, $B$ be finite linear orderings that
   $C$ embeds into. We may suppose that $C = A\cap B$. Then we define an
   ordering on $D = A\cup B$. For $d,e \in D$, let $d\le_D e$ if one of the
@@ -48,7 +49,7 @@
 
 	  \draw[->,thick] (-6, 2)--(0,2) node[right]{$A$};
 	  \foreach \x in {0,...,7}
-		\node at ({-6 + 6/9 * (\x+1)},2)[rectangle,fill=magenta, inner sep=2pt]{};
+		\node at ({-6 + 6/9 * (\x+1)},2)[rectangle,fill=red, inner sep=2pt]{};
 	  \foreach \x in {2,5,6}
 		\node at ({-6 + 6/9 * (\x+1)},2)[circle,fill=green, inner sep=2pt]{};
 
@@ -68,7 +69,7 @@
 	  \foreach \x in {3,7,9}
 		\node at ({-5+10/13*(\x+1)}, 4)[circle,fill=green,inner sep=2pt]{};
 	  \foreach \x in {0,1,4,5,10}
-		\node at ({-5+10/13*(\x+1)}, 4)[rectangle,fill=magenta,inner sep=2pt]{};
+		\node at ({-5+10/13*(\x+1)}, 4)[rectangle,fill=red,inner sep=2pt]{};
 	  \foreach \x in {2,6,8,11}
 		\node at ({-5+10/13*(\x+1)}, 4)[star,fill=blue,inner sep=2pt]{};
 	\end{tikzpicture}
@@ -77,14 +78,14 @@
   \vspace{0.5cm}
 
   On the other hand $\cL$ cannot have $WHP$. This follows from the fact that
-  that only automoprhism of a finite linear ordering is identity, so 
-  we cannot extend a partial automoprhism sending exactly one element to
-  some distinct element. However, in this case, generic automoprhism exists
+  that only automorphism of a finite linear ordering is identity, so 
+  we cannot extend a partial automorphism sending exactly one element to
+  some distinct element. However, in this case, generic automorphism exists
   which was shown by Truss \cite{truss_gen_aut}.
 
   \begin{definition}
-	Let $X$ be a set. A ternarny relation $\le^C \subseteq X^3$ is a
-	\emph{cyclic order}, where we denote $(a,b,c)\in\le^C$ as $a\le^{C}_{b}c$
+	Let $X$ be a set. A ternary relation $\le^C \subseteq X^3$ is a
+	\emph{cyclic order}, where we denote $(a,b,c)\in{\le^C}$ as $a\le^{C}_{b}c$
 	(or simply $a\le_bc$ when there's only one relation in the context),
 	when it satisfies the following properties:
 	\begin{itemize}
@@ -96,7 +97,7 @@
 	\end{itemize}
   \end{definition}
 
-  It is easy to visualise a cyclic ordering as a directed (\textit{nomen omen})
+  It is easy to visualize a cyclic ordering as a directed (\textit{nomen omen})
   cycle. For example, a 11-element cyclic order could be drawn like this:
 
   \begin{figure}[h]
@@ -133,24 +134,37 @@
   the linear orders. The Fraïssé limit of $\cC$ is a countable unit circle.
 
   $\cC$ hasn't WHP by the similar argument to this for linear
-  orderings. Imagine a cycling order of three elements and a partial automoprhism
+  orderings. Imagine a cycling order of three elements and a partial automorphism
   with one fixed point and moving second element to the third. This cannot be
-  extended to automoprhism of any finite cyclic order.
+  extended to automorphism of any finite cyclic order.
 
   Also, $\cC$ cannot have CAP. A reason to that is that it do not admit
   canonical amalgamation over the empty structure see this by taking 
-  1-element cyclic order and 3-element cyclic order with automoprhism other 
+  1-element cyclic order and 3-element cyclic order with automorphism other 
   than identity).
 
- %  \begin{example}
-	% The class of all finitely generated vector spaces over a countable field
-	% $\cV$ is a Fraïssé class with WHP and CAP.
- %  \end{example}
- %
- %  The prove of this is relatively easy knowing that there is essentially one
- %  vector space of finite dimension and that every linear independent subset
- %  of a vector space can be extended to a basis of this space. The Fraïssé limit
- %  of $\cV$ is the $\omega$-dimensional vector space.
+  In contrast to linear orderings the Fraïssé limit $\Sigma = \Flim{\cC}$
+  has no generic automorphism. Consider the set $A$ of automorphisms of $\Sigma$
+  with at least one finite orbit of size greater than $1$. It is open, not dense
+  and closed on conjugation. Openness follows from the fact that all
+  finite orbits of a given automorphism have the same size. Thus $A$ can be
+  represented as a union of basic set generated by finite cycles of 
+  length greater than $1$. It is not dense, as it has empty intersection with
+  basic set generated by identity of a single element. It is also closed on
+  taking conjugation, as the order of elements does not change when conjugating.
+  Thus there cannot be a dense conjugacy class in $\Aut(\Sigma)$ and so there's
+  no generic automorphism.
+
+  \begin{example}
+	The class of all finitely generated vector spaces over a countable field
+	$\cV$ is a Fraïssé class with WHP and CAP.
+  \end{example}
+
+  The prove of this is relatively easy, knowing that there is essentially one
+  vector space of every finite dimension and that every linear independent subset
+  of a vector space can be extended to a basis of this space. The Fraïssé limit
+  of $\cV$ is the $\omega$-dimensional vector space. Thus, by our key Theorem
+  \ref{theorem:key-theorem} we know that it has a generic automorphism.
 
   \begin{example}
 	The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free
@@ -158,12 +172,16 @@
   \end{example}
 
   We have already shown this fact. Thus get that the random graph has a generic
-  automoprhism.
+  automorphism.
 
   \begin{example}
-	Graphs without triangles.
-  \end{example}
-  \begin{example}
-	Graphs without 3-paths.
+	A $K_n$-free graph is a graph with no $n$-clique as its subgraph. 
+	Let $\cG_n$ be the class of finite \emph{$K_n$-free} graphs. $\cG_n$ 
+	is a Fraïssé class with WHP and free amalgamation.
   \end{example}
+
+  Showing that $\cG_n$ is indeed a Fraïssé class is almost the same as in
+  normal graphs, together with free amalgamation. WHP is trickier and the proof
+  can be seen in \cite{eppa_presentation} Theorem 3.6. Hence, $\Flim(\cG_n)$
+  has a generic automorphism.
 \end{document}
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