From 37bb95dea0ac896d9c409fe64f3c1ce99ca43bb2 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 24 Aug 2022 23:22:12 +0200
Subject: Fixes
---
sections/examples.tex | 63 ++++++++++++++++++++++++++++-----------------------
1 file changed, 35 insertions(+), 28 deletions(-)
(limited to 'sections/examples.tex')
diff --git a/sections/examples.tex b/sections/examples.tex
index fdb8c6a..3d276ff 100644
--- a/sections/examples.tex
+++ b/sections/examples.tex
@@ -1,6 +1,41 @@
\documentclass[../lic_malinka.tex]{subfiles}
\begin{document}
+ In this section we give examples and anti-examples of Fraïssé classes
+ with WHP or CAP.
+
+
+ \begin{example}
+ The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free
+ amalgamation.
+ \end{example}
+
+ We have already shown this fact. Thus get that the random graph has a generic
+ automorphism.
+
+ \begin{example}
+ 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.
+
+ \begin{example}
+ The class $\cV$ of all finitely generated vector spaces over a countable field
+ is a Fraïssé class with WHP and CAP.
+ \end{example}
+
+ Vector spaces of the same dimension are isomorphic, thus it is obvious that
+ $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can
+ always embed space with smaller dimension into the bigger one. Amalgamation
+ works exactly the same. In fact, such amalgamation is indeed canonical.
+
+ Now we give some anti-examples:
+
\begin{example}
Let $\cL$ be the class of all finite linear orderings. Then:
\begin{enumerate}
@@ -155,32 +190,4 @@
Thus there cannot be a dense conjugacy class in $\Aut(\Sigma)$ and so there's
no generic automorphism.
- \begin{example}
- The class $\cV$ of all finitely generated vector spaces over a countable field
- is a Fraïssé class with WHP and CAP.
- \end{example}
-
- Vector spaces of the same dimension are isomorphic, thus it is obvious that
- $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can
- always embed space with smaller dimension into the bigger one. Amalgamation
- works exactly the same. In fact, such amalgamation is indeed canonical.
-
- \begin{example}
- The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free
- amalgamation.
- \end{example}
-
- We have already shown this fact. Thus get that the random graph has a generic
- automorphism.
-
- \begin{example}
- 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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