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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 $A = \bigcup_{n\in\bN}A_n$, where $A_n$ are nowhere dense subsets of $X$ (i.e. $\Int(\bar{A_n}) = \emptyset$). 
    \end{definition}

    \begin{definition}
        We say that $A$ is \emph{comeagre} in $X$ if it is a complement of a meager set. Equivalently, a set is comeagre iff it contains a countable intersection of open dense sets.
    \end{definition}
    
    % \begin{example}
    Every countable set is nowhere dense in any $T_1$ space, so, for example, $\bQ$ is meager in $\bR$ (though being dense), which means that the set of irrationals is comeagre. Another example is...
    % \end{example}

    \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 $P$ \emph{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{definition}
        Let $X$ be a nonempty topological space and let $A\subseteq X$. The \emph{Banach-Mazur game of $A$}, denoted as $G^{\star\star}(A)$ is defined as follows: Players $I$ and $II$ take turns in playing nonempty open sets $U_0, V_0, U_1, V_1,\ldots$ such that $U_0 \supseteq V_0 \supseteq U_1 \supseteq V_1 \supseteq\ldots$. We say that player $II$ wins the game if $\bigcap_{n}V_n \subseteq A$.
    \end{definition}

    There is an important theorem on the Banach-Mazur game: $A$ is comeagre
    iff $II$ can always choose sets $V_0, V_1, \ldots$ such that it wins. Before we prove it we need to define notions necessary to formalize this theorem.
    
    \begin{definition}
        $T$ is \emph{the tree of all legal positions} in the Banach-Mazur game $G^{\star\star}(A)$ when $T$ consists of all finite sequences $(W_0, W_1,\ldots, W_n)$, where $W_i$ are nonempty open sets such that $W_0\supseteq W_1\supseteq\ldots\supseteq W_n$. In another words, $T$ is a pruned tree on $\{W\subseteq X\mid W \textrm{is open nonempty}\}$. 
    
        By $[T]$ we denote the set of all "infinite branches" of $T$, i.e. infinite sequences $(U_0, V_0, \ldots)$ such that $(U_0, V_0, \ldots U_n, V_n)\in T$ for any $n\in \bN$.
    \end{definition}


    \begin{definition}
        A \emph{strategy} for $II$ in $G^{\star\star}(A)$ is a subtree $\sigma\subseteq T$ such that
        \begin{enumerate}[label=(\roman*)]
            \item $\sigma$ is nonempty,
            \item if $(U_0, V_0, \ldots, V_n)\in\sigma$, then for all open nonempty $U_{n+1}\subseteq V_n$, $(U_0, V_0, \ldots, V_n, U_{n+1})\in\sigma$,
            \item if $(U_0, V_0, \ldots, U_{n})\in\sigma$, then for unique $V_n$, $(U_0, V_0, \ldots,  U_{n}, V_n)\in\sigma$. 
        \end{enumerate}
    \end{definition}

    Intuitively, the strategy $\sigma$ works as follows: $I$ starts playing $U_0$ as any open subset of $X$, then $II$ plays unique (by (iii)) $V_0$ such that $(U_0, V_0)\in\sigma$. Then $I$ responds by playing any $U_1\subseteq V_0$ and $II$ plays uniqe $V_1$ such that $(U_0, V_0, U_1, V_1)\in\sigma$, etc.

    \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}