From 14916d95484bdac8fa95d66ca8a7fb37d69efee6 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Mon, 28 Mar 2022 23:57:24 +0200
Subject: dokonczony dowod

---
 lic_malinka.pdf | Bin 372613 -> 374634 bytes
 lic_malinka.tex |  31 ++++++++++++++++++++++++-------
 2 files changed, 24 insertions(+), 7 deletions(-)

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 751fe33..8229069 100644
Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ
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}
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