dokonczony dowod

1 files changed, 24 insertions, 7 deletions

diff --git a/lic_malinka.tex b/lic_malinka.tex
index a7ead6d..f7a3570 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -452,7 +452,7 @@
   \end{lemma}

 

   This lemma will play a major role in the later parts of the paper. Weak

-  ultrahomogenity is an easier and more intuitive property and it will prove

+  ultrahomogeneity is an easier and more intuitive property and it will prove

   useful when recursively constructing the generic automorphism of a Fraïssé 

   limit.

 

@@ -507,7 +507,7 @@
     Wihtout loss of generality assume that this embedding is simply inclusion. 

     Let $f$ be the partial isomorphism from $X\sqcup Y$ to $H$, with $X$ and

     $Y$ projected to the part of $H$ that come from $X$ and $Y$ respectively. 

-    By the ultrahomogenity of $\FrGr$ this isomoprhism extends to an automorphism

+    By the ultrahomogeneity of $\FrGr$ this isomoprhism extends to an automorphism

     $\sigma\in\Aut(\FrGr)$. Then $v = \sigma^{-1}(w)$ is the vertex we sought. 

   \end{proof}

 

@@ -542,6 +542,7 @@
   manner is in fact isomorphic to the random graph $\FrGr$.

 

   \begin{proposition}

+    \label{proposition:finite-graphs-whp}

     The class of finite graphs $\cG$ has the weak Hrushovski property. 

   \end{proposition}

 

@@ -591,15 +592,31 @@
 

   \begin{proposition}

     The Fraïssé limit of $\cH$ interpreted as a plain graph is isomorphic to

-    the random graph $\FrGr$.

+    the random graph $\FrGr$. 

   \end{proposition}

 

   \begin{proof}

     It is enough to show that $\FrAut = \Flim(\cH)$ has the random graph

-    property. Take any finite disjoint $X, Y\subseteq \FrAut$. Let $G_{XY}$ be

-    the graph induced by $X\cup Y$. Take $H=G_{XY}\sqcup {v}$ as a supergraph

-    of $G_{XY}$ with one new vertex $v$ connected to all verticies of $X$ and

-    to none of $Y$. By the weak homogeneity $H$ embedds to $\FrAut$

+    property. Take any finite disjoint $X, Y\subseteq \FrAut$. Without the loss

+    of generality assume that $X\cup Y$ is invariant to $\sigma$, i.e.

+    $\sigma(v)\in X\cup Y$ for $v\in X\cup Y$. This assumption can be done

+    because there are no infinite orbits in $\sigma$, which in turn is due to

+    the fact that $\cH$ is the age of $\FrAut$. 

+

+    Let $G_{XY}$ be the graph induced by $X\cup Y$. Take $H=G_{XY}\sqcup {v}$ 

+    as a supergraph of $G_{XY}$ with one new vertex $v$ connected to all 

+    verticies of $X$ and to none of $Y$. By the proposition

+    \ref{proposition:finite-graphs-whp} we can extend $H$ together with its

+    partial isomoprhism $\sigma\upharpoonright_{X\cup Y}$ to a graph $R$ with

+    automorphism $\tau$. Once again, without the loss of generality we can

+    assume that $R\subseteq\FrAut$, because $\cH$ is the age of $\FrAut$. But

+    $R\upharpoonright_{G_{XY}}$ together with $\tau\upharpoonright_{G_{XY}}$

+    are isomorphic to the $G_{XY}$ with $\sigma\upharpoonright_{G_{XY}}$.

+

+    Thus, by ultrahomogeneity of $\FrAut$ this isomorphism extends to an 

+    automorphism $\theta$ of $(\FrAut, \sigma)$. Then $\theta(v)$ is the vertex

+    that is connected to all verticies of $X$ and none of $Y$, because

+    $\theta[R\upharpoonright_X] = X, \theta[R\upharpoonright_Y] = Y$.

   \end{proof}

 

   \section{Conjugacy classes in automorphism groups}