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+\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}