More examples

4 files changed, 64 insertions, 34 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index d5a991d..067df4a 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
diff --git a/lic_malinka.tex b/lic_malinka.tex
index 47e1499..ed73de7 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -29,12 +29,18 @@
 \usepackage{etoolbox}

 

 \usepackage{xcolor} 

-\definecolor{green}{RGB}{0,127,0}

-\definecolor{redd}{RGB}{191,0,0} 

-

-\definecolor{red}{RGB}{105,89,205}

 \usepackage[colorlinks=true]{hyperref}

-

+\hypersetup{

+	colorlinks,

+	linkcolor={red!50!black},

+	citecolor={blue!50!black},

+	urlcolor={blue!80!black}

+}

+% \definecolor{green}{RGB}{0,127,0}

+% \definecolor{redd}{RGB}{191,0,0} 

+%

+% \definecolor{red}{RGB}{105,89,205}

+%

 \usepackage[notref, notcite]{showkeys} 

 \usepackage[cmtip,arrow]{xy}

 

@@ -62,6 +68,7 @@
 \newcommand{\bQ}{\mathbb Q}

 \newcommand{\cK}{\mathcal K}

 \newcommand{\cL}{\mathcal L}

+\newcommand{\cT}{\mathcal T}

 

 \newcommand{\FrAut}{\Pi}

 \newcommand{\FrGr}{\Gamma}

diff --git a/licmalinka.bib b/licmalinka.bib
index 26dd13f..051ae83 100644
--- a/licmalinka.bib
+++ b/licmalinka.bib
@@ -73,5 +73,10 @@
   url={https://core.ac.uk/download/pdf/83934818.pdf},
   year={2017},
   publisher={The University of Leeds (Department of Pure Mathematics)}
+},
+@article{eppa_presentation,
+  title={Fraïssé Limits, Hrushovski Property and Generic Automorphisms},
+  author={Shixiao Liu},
+  url={https://logic.pku.edu.cn/ann_attachments/eppa-presentation.pdf},
+  year={2017}
 }
-
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}