Main proof maybe finished

1 files changed, 66 insertions, 12 deletions

diff --git a/lic_malinka.tex b/lic_malinka.tex
index 47dea88..85541cf 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -717,7 +717,8 @@
 	Our goal is to choose $g_i$ in such a manner that the resulting function

 	$g = \bigcap^{\infty}_{i=0}g_i$ will be an automorphism of the random graph

 	such that $(\FrGr, g) = \Flim{\cH}$, i.e. the Fraïssé limit of finite

-	graphs with automorphism. By the Fraïssé theorem \ref{theorem:fraisse_thm}

+	graphs with automorphism. Precisely, $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$.

+	By the Fraïssé theorem \ref{theorem:fraisse_thm}

 	it will follow that $(\FrGr, g)\cong (\FrAut, \sigma)$. By the 

 	proposition \ref{proposition:graph-aut-is-normal} we can assume without

 	the loss of generatlity that $\FrAut = \FrGr$ as a plain graph. Hence,

@@ -732,11 +733,18 @@
 	i.e. $(A_n, \alpha_n)\subseteq (B_n, \beta_n)$. Also, it may be that 

 	$A_n$ is an empty graph. We enumerate the verticies of the random graph 

 	$\FrGr = \{v_0, v_1, \ldots\}$.

-	

-	Just for sake of fixing a technical problem, let $g_{-1} = \emptyset$. 

-	Suppose that player \textit{I} in the $n$-th move chose a finite partial

-	isomoprhism $f_n$. We will construct $g_n\supseteq f_n$ such that

-	following properties hold:

+

+	Fix a bijection $\gamma\colon\bN\times\bN\to\bN$ such that for any

+	$n, m\in\bN$ we have $\gamma(n, m) \ge n$. This bijection naturaly induces

+	a well ordering on $\bN\times\bN$. This will prove useful later, as the

+	main argument of the proof will be constructed as a bookkeeping argument.

+

+	Just for sake of fixing a technical problem, let $g_{-1} = \emptyset$ and

+	$X_{-1} = \emptyset$. 

+	Suppose that player \textit{I} in the $n$-th move chooses a finite partial

+	isomoprhism $f_n$. We will construct $g_n\supseteq f_n$ and a set $X_n\subseteq\bN^2$

+	such that following properties hold:

+

 	\begin{enumerate}[label=(\roman*)]

 	  \item $g_n$ is an automorphism of the induced subgraph $\FrGr_n$,

 	  \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined,

@@ -747,14 +755,15 @@
 		$(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$

 		is an embedding	of $(A_{n,k}, \alpha_{n,k})$ in the $\FrGr_{n-1}$ (which

 		is the graph induced by $g_{n-1}$). Let 

-		$(i, j) = \min\{\{0, 1, \ldots\} \times \bN \setminus X_{n-1}\}$. Then

+		$(i, j) = \min\{(\{0, 1, \ldots\} \times \bN) \setminus X_{n-1}\}$ (with the

+		order induced by $\gamma$). Then $X_n = X_{n-1}\cup\{(i,j)\}$ and

 		$(B_{n,k}, \beta_{n,k})$ embedds in $(\FrGr_n, g_n)$ so that this diagram

 		commutes:

 

 		\begin{center}

 		  \begin{tikzcd}

-			(A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "f_{i,j}"] & (\FrGr_n, g_n) \\

-			(B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\hat{f}_{i,j}"']

+			(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}

@@ -762,10 +771,55 @@
 	First item makes sure that no inifite orbit will not be present in $g$. The 

 	second item together with the first one are necessary for $g$ to be an 

 	automorphism of $\FrGr$. The third item is the one that gives weak

-	ultrahomogeneity. 

+	ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed.

+	

+	First, we will suffice the (iii) item. Namely, we will construct $\FrGr'_n, g'_n$

+	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 embedds 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 recurisve assumption $\FrGr_{n-1}\subseteq\FrGr$. The 

+	$\FrGr'_n \setminus\FrGr_{n-1} = B_{i,j}\setminus A_{i,j}$ can be found in

+	$\FrGr$ by the random graph property -- we can find verticies of the remaining

+	part of $B_{i,j}$ each at a time so that all edges are correct.

+

+	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

+	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.

+	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.

+

+	Take any finite graph with automorphism $(B, \beta)$. Then, there are

+	$i, j$ 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 $(B, \beta)$ embedds into $(\FrGr_n, g_n)$, hence it embedds

+	into $(\FrGr, g)$. With a similar argument we can see that $(\FrGr, g)$ is

+	weakly ultrahomogenous.

+

+	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.

+	possible $g$'s we end up with) is comeagre in $G$, thus $\sigma^G$ is also

+	comeagre and $\sigma$ is a generic automorphism.

   \end{proof}

 

-

-

   \printbibliography

 \end{document}