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\title{Tytuł}
\author{Franciszek Malinka}
\begin{document}
\begin{abstract}
Abstract
\end{abstract}
\section{Introduction}
\section{Preliminaries}
\subsection{Descriptive set theory}
\begin{definition}
Suppose $X$ is a topological space and $A\subseteq X$. We say that $A$ is \emph{meagre} in $X$ if... Wea say that $A$ is \emph{comeagre} in $X$ if... .
\end{definition}
\begin{definition}
We say that a topological space $X$ is a \emph{Baire space} if every comeagre subset of $X$ is dense in $X$ (equivalently, every meagre set has empty interior).
\end{definition}
\begin{definition}
Suppose $X$ is a Baire space. We say that a property $P$ holds generically for a point in $x\in X$ if $\{x\in X\mid P\textrm{ holds for }x\}$ is comeagre in $X$.
\end{definition}
\begin{example}
content
\end{example}
\subsection{Fraïssé classes}
\begin{fact}[Fraïssé theorem]
\label{fact:fraisse_thm}
Suppose $\cC$ is a class of finitely generated $L$-structures such that...
Then there exists a unique up to isomorphism counable $L$-structure $M$ such that...
\end{fact}
\begin{definition}
For $\cC$, $M$ as in Fact~\ref{fact:fraisse_thm}, we write $\Flim(\cC)\coloneqq M$.
\end{definition}
\begin{fact}
If $\cC$ is a uniformly locally finite Fraïssé class, then $\Flim(\cC)$ is $\aleph_0$-categorical and has quantifier elimination.
\end{fact}
\section{Conjugacy classes in automorphism groups}
\subsection{Prototype: pure set}
In this section, $M=(M,=)$ is an infinite countable set (with no structure beyond equality).
\begin{proposition}
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 $n$.
\end{proposition}
\begin{proposition}
The conjugacy class of $f\in \Aut(M)$ is dense if and only if...
\end{proposition}
\begin{proposition}
If $f\in \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is meagre.
\end{proposition}
\begin{proposition}
An automorphism $f$ of $M$ is generic if and only if...
\end{proposition}
\begin{proof}
\end{proof}
\subsection{More general structures}
\begin{proposition}
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 $(M,f_1)\cong (M,f_2)$.
\end{proposition}
\begin{definition}
We say that a Fraïssé class $\cC$ has \emph{weak Hrushovski property} (\emph{WHP}) if for every $A\in \cC$ and partial automorphism $p\colon A\to A$, there is some $B\in \cC$ 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,"\bar p"]&B\\
A\ar[u,"i"]\ar[r,"p"]&A\ar[u,"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 orbits of $f$ are finite.
\end{proposition}
\begin{proposition}
Suppose $\cC$ is a Fraïssé class in an arbitrary countable language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$ ...
\end{proposition}
\subsection{Random graph}
\begin{definition}
The \emph{random graph} is...
\end{definition}
\begin{fact}
The
\end{fact}
\begin{proposition}
Generically, the set of fixed points of $f\in \Aut(M)$ is isomorphic to $M$ (as a graph).
\end{proposition}
\end{document}
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