Main proof

1 files changed, 100 insertions, 16 deletions

diff --git a/lic_malinka.tex b/lic_malinka.tex
index f7a3570..47dea88 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -1,10 +1,10 @@
 \documentclass[11pt, a4paper, final]{amsart} 

 \setlength{\emergencystretch}{2em}

 

+\usepackage[utf8]{inputenc}

 \usepackage[backend=biber]{biblatex}

 \addbibresource{licmalinka.bib}

 

-

 \usepackage[T1]{fontenc} 

 \usepackage{mathtools}

 \usepackage[activate={true,nocompatibility},final,tracking=true,kerning=true,spacing=true,stretch=10,shrink=10]{microtype}

@@ -19,7 +19,6 @@
 \usepackage[charter, expert, greekuppercase=italicized, greekfamily=didot]{mathdesign}

 \usepackage{mathtools} 

 \usepackage{enumitem} 

-\usepackage[utf8]{inputenc}

 \usepackage{tikz-cd} 

 \usepackage{tikz}

 \usetikzlibrary{arrows,arrows.meta}

@@ -541,6 +540,20 @@
   Using this fact one can show that the graph constructed in the probabilistic

   manner is in fact isomorphic to the random graph $\FrGr$.

 

+  \begin{definition} We say that a Fraïssé class $\bK$ has \emph{weak

+    Hrushovski property} (\emph{WHP}) if for every $A\in \bK$ and an isomorphism

+    of substructures of $A$ $p\colon A\to A$, there is some $B\in \bK$ such 

+    that $p$ can be extended to an automorphism of $B$, i.e.\ there is an 

+    embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following

+    diagram commutes: 

+    \begin{center} 

+      \begin{tikzcd} 

+        B\ar[r,dashed,"\bar p"]&B\\

+        A\ar[u,dashed,"i"]\ar[r,"p"]&A\ar[u,dashed,"i"] 

+      \end{tikzcd} 

+    \end{center}

+  \end{definition}

+

   \begin{proposition}

     \label{proposition:finite-graphs-whp}

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

@@ -568,7 +581,7 @@
 

     Take any graphs $(A, \alpha), (B, \beta), (C,\gamma)$ such that $A$ embedds

     into $B$ and $C$. Let $D$ be the amalgamation of $B$ and $C$ over $A$ as in

-    the proof for the plain graphs. We will define the automorphis

+    the proof for the plain graphs. We will define the automorphism

     $\delta\in\Aut(D)$ so it extends $\beta$ and $\gamma$. (TODO: chyba nie

     tylko extends ale coś więcej: wiem o co chodzi, ale nie wiem jak to

     napisać) We let $\delta_{\upharpoonright X} = \id_X$ for $X\in \{A,

@@ -591,6 +604,7 @@
   surprisingly, $\FrAut$ is in fact a random graph.

 

   \begin{proposition}

+	\label{proposition:graph-aut-is-normal}

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

     the random graph $\FrGr$. 

   \end{proposition}

@@ -647,6 +661,7 @@
   \subsection{More general structures}

 

   \begin{fact} 

+	\label{fact:conjugacy}

     Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. 

     Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong

     (M,f_2)$ as structures with one additional unary relation that is

@@ -660,19 +675,6 @@
     $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate.

   \end{proof} 

 

-  \begin{definition} We say that a Fraïssé class $\bK$ has \emph{weak

-    Hrushovski property} (\emph{WHP}) if for every $A\in \bK$ and an isomorphism

-    of substructures of $A$ $p\colon A\to A$, there is some $B\in \bK$ such 

-    that $p$ can be extended to an automorphism of $B$, i.e.\ there is an 

-    embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following

-    diagram commutes: 

-    \begin{center} 

-      \begin{tikzcd} 

-        B\ar[r,dashed,"\bar p"]&B\\

-        A\ar[u,dashed,"i"]\ar[r,"p"]&A\ar[u,dashed,"i"] 

-      \end{tikzcd} 

-    \end{center}

-  \end{definition}

 

   % \begin{proposition} Suppose $\cC$ is a Fraïssé class in a relational

   % language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$, all

@@ -683,5 +685,87 @@
     %   \begin{proposition} Generically, the set of fixed points of $f\in

   %   \Aut(M)$ is isomorphic to $M$ (as a graph).  \end{proposition}

 

+  \begin{theorem}

+	\label{theorem:generic_aut_graph}

+	Let $\FrGr$ be the Fraïssé limit of the class of all finite graphs $\bK$.

+	Then $\FrGr$ has a generic automorphism $\tau\in\Aut(\FrGr)$, i.e. the

+	conjugacy class of $\tau$ is comeagre in $G = \Aut(\FrGr)$.

+  \end{theorem}

+

+  \begin{proof}

+	We will construct a strategy for the second player in the Banach-Mazur game

+	on the topological space $G$. This strategy will give us a subset 

+	$A\subseteq G$ and as we will see, this will also be a subset of the 

+	conjugacy class of $\tau$. By the Banach-Mazur theorem 

+	\ref{theorem:banach_mazur_thm} this will prove that the class is comeagre.

+

+	Recall, $G$ has a basis consisting of open

+	sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some

+	finite set $A\subseteq \FrGr$ and some automorphism $g_0\in G$. In other

+	words, a basic open set is a set of all extensions of some finite partial

+	isomorphism $g_0$ of $\FrGr$. By $B_{g}\subseteq G$ we denote a basic 

+	open subset given by a finite partial isomorphism $g$. From now on we will

+	consider only finite partial isomorphism $g$ such that $B_g$ is nonempty.

+

+	With the use of corollary \ref{corollary:banach-mazur-basis} we can consider 

+	only games, where both players choose finite partial isomorphisms. Namely,

+	player \textit{I} picks functions $f_0, f_1,\ldots$ and player \textit{II}

+	chooses $g_0, g_1,\ldots$ such that

+	$f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify

+	the coresponding basic open subsets $B_{f_0}\supseteq B_{g_0}\supseteq\ldots$.

+

+	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}

+	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,

+	by the fact	\ref{fact:conjugacy}, $g$ and $\sigma$ conjugate in $G$.

+

+	Once again, by the Fraïssé theorem \ref{theorem:fraisse_thm} and the

+	\ref{lemma:weak_ultrahom} lemma constructing $g_i$'s in a way such that

+	age of $(\FrGr, g)$ is exactly $\cH$ and so that it is weakly ultrahomogenous

+	will produce expected result. First, let us enumerate all pairs of finite

+	graphs with automorphism $\{\langle(A_n, \alpha_n), (B_n, \beta_n)\rangle\}_{n\in\bN}$

+	such that the first element of the pair embedds by inclusion in the second,

+	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:

+	\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,

+	  \item let 

+		$\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$

+		be the enumeration of all pairs of finite graphs with automorphism such

+		that the first is a substructure of the second, i.e. 

+		$(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, 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

+		$(B_{n,k}, \beta_{n,k})$ 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}"']

+		  \end{tikzcd}

+		\end{center}

+	\end{enumerate}

+

+	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. 

+  \end{proof}

+

+

+

   \printbibliography

 \end{document}