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Diffstat (limited to 'sections/fraisse_classes.tex')
| -rw-r--r-- | sections/fraisse_classes.tex | 9 |
1 files changed, 5 insertions, 4 deletions
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex index 9110f44..dc6392d 100644 --- a/sections/fraisse_classes.tex +++ b/sections/fraisse_classes.tex @@ -149,6 +149,7 @@ \end{proof} \begin{definition} + \label{definition:random_graph} The \emph{random graph} is the Fraïssé limit of the class of finite graphs $\cG$ denoted by $\FrGr = \Flim(\cG)$. \end{definition} @@ -253,7 +254,7 @@ \end{tikzcd} \end{center} - We have deliberately omited names for embeddings of $C$. Of course, + We have deliberately omitted names for embeddings of $C$. Of course, the functor has to take them into account, but this abuse of notation is convenient and should not lead into confusion. \item Let $A\leftarrow C\rightarrow B$, $A'\leftarrow C\rightarrow B'$ be cospans @@ -414,13 +415,13 @@ theorem \ref{theorem:fraisse_thm} it suffices to show that the age of $\Pi$ is $\cC$ and that it has the weak ultrahomogeneity in the class $\cC$. The former comes easily, as for every structure $A\in \cC$ we have the structure - $(A, \id_A)\in \cD$, which means that the structure $A$ embedds into $\Pi$. - Also, if a structure $(B, \beta)\in\cD$ embedds into $\cD$, then $B\in\cC$. + $(A, \id_A)\in \cD$, which means that the structure $A$ embeds into $\Pi$. + Also, if a structure $(B, \beta)\in\cD$ embeds into $\cD$, then $B\in\cC$. Hence, $\cC$ is indeed the age of $\Pi$. Now, take any structure $A, B\in\cC$ such that $A\subseteq B$. Without the loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}$ be the - smallest structure closed on the automorphism $\sigma$ and containg $A$. + smallest structure closed on the automorphism $\sigma$ and containing $A$. It is finite, as $\cC$ is the age of $\Pi$. By the weak Hrushovski property, of $\cC$ let $(\bar{B}, \beta)$ be a structure extending $(B\cup \bar{A}, \sigma\upharpoonright_{\bar{A}})$. Again, we may assume |