Let $\mathfrak{A}$ be a first-order structure over a relational language and let $\kappa$ be an infinite cardinal. Lets say that $\mathfrak{A}$ has the $\kappa$-*property* if for every structure $\mathfrak{B}$ of size $\kappa$ over the same language as $\mathfrak{A}$ we have: $$\mathfrak{B} \mbox{ embeds into }\mathfrak{A} \Longleftrightarrow \mbox{ Every finite } \mathfrak{B}\_0\subseteq\mathfrak{B} \mbox{ embeds into } \mathfrak{A}.$$


Has this property been studied before? If so, how is it called?


Note that if the theory of $\mathfrak{A}$ is $|\mathfrak{A}|$-categorical, then a simple compactness argument shows that $\mathfrak{A}$ has the $\kappa$-property for every $\kappa\leq|\mathfrak{A}|$. I don´t expect the other implication to be true, but no examples come to my mind. Hence the question in the title.


**Edit:** I apologize because I have broken two of the rules for [good MO questions](https://meta.mathoverflow.net/questions/882/how-to-write-a-good-mathoverflow-question?cb=1). One is *Make your title your question* and the other is *Do your homework*. Thanks to Joel, Andreas and Noah for their examples, I should really have put more effort into finding them. But what I´m really interested in is the question in the body:


**Has this property been studied before? If so, how is it called?**



Does this property of a first-order structure imply categoricity?
The answer is no for uncountable cardinals $\kappa$. Let $\mathfrak{A}=\langle A,U\rangle$ be a set $A$ of size $\kappa$ with a unary predicate $U\subset A$, where $U$ and $A-U$ both have size $\kappa$. It follows that $\mathfrak{A}$ has your property, since every structure $\mathfrak{B}=\langle B,U^B\rangle$ of size at most $\kappa$ in that language embeds into $\mathfrak{A}$. But the theory of $\mathfrak{A}$ is the theory of an infinite/co-infinite predicate, which is complete because it is countably categorical, but which is not uncountably categorical because there are models of uncountable size $\kappa$ where the predicate is interpreted in a set of size less than $\kappa$, and such a model is not isomorphic to $\mathfrak{A}$. 


