\documentclass[11pt, a4paper, final]{amsart}
\setlength{\emergencystretch}{2em}


\usepackage[T1]{fontenc}
\usepackage{mathtools}
\usepackage[activate={true,nocompatibility},final,tracking=true,kerning=true,spacing=true,stretch=10,shrink=10]{microtype}
\microtypecontext{spacing=nonfrench}
\usepackage[utf8]{inputenc}

\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{XCharter}
\usepackage[charter,expert, greekuppercase=italicized, greekfamily=didot]{mathdesign}
\usepackage{mathtools}
\usepackage{enumitem}
\usepackage[utf8]{inputenc}
\usepackage{tikz-cd}
\usepackage{tikz}

\usepackage{etoolbox}


\usepackage{xcolor}
\definecolor{green}{RGB}{0,127,0}
\definecolor{redd}{RGB}{191,0,0}
\definecolor{red}{RGB}{105,89,205}
\usepackage[colorlinks=true]{hyperref}

\usepackage[notref, notcite]{showkeys}
\usepackage[cmtip,arrow]{xy}
%\usepackage[backend=biber,
%url=false,
%isbn=false,
%backref=true,
%citestyle=alphabetic,
%bibstyle=alphabetic,
%autocite=inline,
%maxnames=99,
%minalphanames=4,
%maxalphanames=4,
%sorting=nyt,]{biblatex}
%\addbibresource{linear_strucures.bib}

\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\st}{st}
\DeclareMathOperator{\Flim}{FLim}

\newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}}
\newcommand{\cC}{\mathcal C}
\newcommand{\bN}{\mathbf N}
\newcommand{\bR}{\mathbf R}
\newcommand{\bZ}{\mathbf Z}

\DeclareMathOperator{\im}{{Im}}
\DeclareMathOperator{\lin}{{Lin}}
\DeclareMathOperator{\Th}{{Th}}


\newtheorem{theorem}{Theorem}
\numberwithin{theorem}{section}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{question}[theorem]{Question}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem*{theorem2}{Theorem}
\newtheorem*{claim2}{Claim}
\newtheorem*{corollary2}{Corollary}
\newtheorem*{question2}{Question}
\newtheorem*{conjecture2}{Conjecture}


\newtheorem{clm}{Claim}
\newtheorem*{clm*}{Claim}


\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem*{definition2}{Definition}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem*{remark2}{Remark}


\AtEndEnvironment{proof}{\setcounter{clm}{0}}
\newenvironment{clmproof}[1][\proofname]{\proof[#1]\renewcommand{\qedsymbol}{$\square$(claim)}}{\endproof}

\newcommand{\xqed}[1]{%
    \leavevmode\unskip\penalty9999 \hbox{}\nobreak\hfill
    \quad\hbox{\ensuremath{#1}}}


\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}
