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\documentclass[../lic_malinka.tex]{subfiles}
\begin{document}
Model theory is a field of mathematics that classifies and constructs
structures with particular properties (particularly those expressible
in first order logic). It describes classical mathematical
objects in a broader context, abstracts their properties and studies
connections between seemingly unrelated structures.
This work studies limits of Fraïssé classes with additional combinatorial
and categorical properties. Fraïssé classes are frequently used in model
theory,
both as a source of examples and to analyse particular ``generic'' structures.
The notion of Fraïssé class and its limit is
due to the French logician Roland Fraïssé. He also introduced the
back-and-forth argument, a fundamental model theoretical method in
construction of elementarily equivalent structures, upon which
Ehrenfeucht-Fraïssé games are based.
The prototypical example for this paper is the random graph
\ref{definition:random_graph} (also known as the
Rado graph), the Fraïssé limit of the class of finite undirected graphs.
It serves as a useful example, gives an intuition of the Fraïssé limits,
weak Hrushovski property and free amalgamation. Perhaps most importantly,
the random graph has a so-called generic automorphism
\ref{definition:generic_automorphism}, which was first proved
by Truss in \cite{truss_gen_aut}, where he also introduced the term.
The key Theorem \ref{theorem:key-theorem}
says that a Fraïssé class with canonical amalgamation and weak Hrushovski
property has a generic automorphism. The fact that such an automorphism
exists in this case follows from the classical results of Ivanov \cite{ivanov_1999}
and Kechris-Rosendal \cite{https://doi.org/10.1112/plms/pdl007}.
In this work we show a new way to construct
a generic automorphism by expanding the structures of the class by a (total)
automorphism and considering limit of such extended Fraïssé class. We achieve
this by using the Banach-Mazur games, a well known method in the descriptive
set theory, which proves useful in the study of comeagre sets.
% Finally, we show how this construction of the generic automorphism can be
% used to deduce some properties of generic automorphisms
% (see \ref{proposition:fixed_points}, (COŚ JESZCE)).
\end{document}
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