From 29fb1dc0cb80c83f071079009ba487720685f05a Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Tue, 3 May 2022 13:12:54 +0200
Subject: poprawki do kluczowego dowodu

---
 lic_malinka.tex | 81 +++++++++++++++++++++++++++++++++------------------------
 1 file changed, 47 insertions(+), 34 deletions(-)

(limited to 'lic_malinka.tex')

diff --git a/lic_malinka.tex b/lic_malinka.tex
index fe992a1..5e52034 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -663,6 +663,7 @@
   beyond equality).
   
   \begin{proposition} 
+	\label{proposition:cojugate-classes}
 	If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are conjugate if and only 
     if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$ and $f_2$ have the same 
     number of orbits of size $n$.
@@ -703,20 +704,11 @@
 	is an open dense set. It is a sum over basic open sets generated by finite
 	permutations with $m$ in their domain. Denseness is also easy to see.
 
-	Finally, we can say that 
+	Finally, by the proposition \ref{proposition:cojugate-classes}, we can say that 
 	$$\sigma^{\Aut(M)}=\bigcap_{n=1}^\infty A_n \cap \bigcap_{m\in M} B_m,$$
 	which concludes the proof.
-	
   \end{proof}
 
-  % \begin{proposition} 
-    % An automorphism $f$ of $M$ is generic if and only if...
-  % \end{proposition}
-
-  % \begin{proof}
-
-  % \end{proof}
-
   \subsection{More general structures}
 
   \begin{fact} 
@@ -821,10 +813,18 @@
 
 		\begin{center}
 		  \begin{tikzcd}
-			(A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "f_{i,j}"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\
-			(B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\hat{f}_{i,j}"'] & (\FrGr_n, g_n)
+									  & (\FrGr_n, g_n) & \\
+			(B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\hat{f}_{i,j}"] &   & (\FrGr_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\
+									  & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "f_{i,j}"'] \arrow[ul, "\subseteq"]
 		  \end{tikzcd}
 		\end{center}
+
+		% \begin{center}
+		%   \begin{tikzcd}
+		% 	(A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "f_{i,j}"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\
+		% 	(B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\hat{f}_{i,j}"'] & (\FrGr_n, g_n)
+		%   \end{tikzcd}
+		% \end{center}
 	\end{enumerate}
 
 	First item makes sure that no infinite orbit will not be present in $g$. The 
@@ -832,23 +832,35 @@
 	automorphism of $\FrGr$. The third item is the one that gives weak
 	ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed.
 	
-	First, we will suffice the (iii) item. Namely, we will construct $\FrGr'_n, g'_n$
+	First, we will suffice the item (iii). Namely, we will construct $\FrGr'_n, g'_n$
 	such that $g_{n-1}\subseteq g'_n$ and $f_{i,j}$ extends to an embedding of
-	$(B_{i,j}, \beta_{i,j})$ to $(\FrGr'_n, g'_n)$. Without the loss of generality
-	we may assume that $f_{i,j}$ is an inclusion and that $A_{i,j} = B_{i,j}\cap\FrGr_{n-1}$.
-	Then let $\FrGr'_n = B_{i,j}\cup\FrGr_{n-1}$ and $g'_n = g_{n-1}\cup \beta_{i,j}$.
-	Then $(B_{i,j}, \beta_{i,j})$ simply embeds by inclusion in $(\FrGr'_n, g'_n)$,
-	i.e. this diagram commutes:
-
-	\begin{center}
-	  \begin{tikzcd}
-		(A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\
-		(B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\subseteq"'] & (\FrGr'_n, g'_n)
-	  \end{tikzcd}
-	\end{center}
-
-	The argument that those are actually embeddings is almost the same as in
-	proof of the amalgamation property of $\cH$.	
+	$(B_{i,j}, \beta_{i,j})$ to $(\FrGr'_n, g'_n)$. But this can be easily
+	done by the fact, that $\cH$ has the amalgamation property. Moreover, without
+	the loss of generality we can assume that all embeddings are inclusions.
+
+    \begin{center}
+      \begin{tikzcd}
+                                  & (\FrGr'_n, g'_n) & \\
+        (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\subseteq"] &   & (\FrGr_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\
+								  & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "\subseteq"'] \arrow[ul, "\subseteq"]
+      \end{tikzcd}
+    \end{center}
+
+	% Without the loss of generality
+	% we may assume that $f_{i,j}$ is an inclusion and that $A_{i,j} = B_{i,j}\cap\FrGr_{n-1}$.
+	% Then let $\FrGr'_n = B_{i,j}\cup\FrGr_{n-1}$ and $g'_n = g_{n-1}\cup \beta_{i,j}$.
+	% Then $(B_{i,j}, \beta_{i,j})$ simply embeds by inclusion in $(\FrGr'_n, g'_n)$,
+	% i.e. this diagram commutes:
+
+	% \begin{center}
+	%   \begin{tikzcd}
+	% 	(A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\
+	% 	(B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\subseteq"'] & (\FrGr'_n, g'_n)
+	%   \end{tikzcd}
+	% \end{center}
+
+	% The argument that those are actually embeddings is almost the same as in
+	% proof of the amalgamation property of $\cH$.	
 
 	It may be also assumed without the loss of generality that $\FrGr'_n\subseteq \FrGr$.
 	Of course by the recursive assumption $\FrGr_{n-1}\subseteq\FrGr$. The 
@@ -858,12 +870,12 @@
 
 	Now, by the WHP of $\bK$ we can extend the graph $\FrGr'_n\cup\{v_n\}$ together
 	with its partial isomorphism $g'_n$ to a graph $\FrGr_n$ together with its
-	automorphism $g_n\supseteq g'_n$, where without the loss of generality we
+	automorphism $g_n\supseteq g'_n$ and without the loss of generality we
 	may assume that $\FrGr_n\subseteq\FrGr$. This way we've constructed $g_n$
 	that has all desired properties.
 
 	Now we need to see that $g = \bigcap^{\infty}_{n=0}g_n$ is indeed an automorphism
-	of $\FrGr$ such that $(\FrGr, g)$ has the age $\cH$ and has weak ultrahomogeneity.
+	of $\FrGr$ such that $(\FrGr, g)$ has the age $\cH$ and is weakly ultrahomogeneous.
 	It is of course an automorphism of $\FrGr$ as it is defined for every $v\in\FrGr$
 	and is a sum of increasing chain of finite automorphisms.
 
@@ -871,13 +883,14 @@
 	$i, j$ such that $(B, \beta) = (B_{i, j}, \beta_{i,j})$ and $A_{i,j}=\emptyset$.
 	By the bookkeeping there was $n$ such that $(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$.
 	This means that $(B, \beta)$ embeds into $(\FrGr_n, g_n)$, hence it embeds
-	into $(\FrGr, g)$. With a similar argument we can see that $(\FrGr, g)$ is
-	weakly ultrahomogeneous.
+	into $(\FrGr, g)$, thus it has age $\cH$. 
+	With a similar argument we can see that $(\FrGr, g)$ is	weakly ultrahomogeneous.
 
 	By this we know that $g$ and $\sigma$ conjugate in $G$. As we stated in the
-	beginning of the proof, the set of possible outcomes of the game (i.e.
+	beginning of the proof, the set $A$ of possible outcomes of the game (i.e.
 	possible $g$'s we end up with) is comeagre in $G$, thus $\sigma^G$ is also
-	comeagre and $\sigma$ is a generic automorphism.
+	comeagre and $\sigma$ is a generic automorphism, as it contains a comeagre
+	set $A$.
   \end{proof}
 
   \printbibliography
-- 
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