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Community Bot
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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 questionsgood MO questions. 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?

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. 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?

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. 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?

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Ramiro de la Vega
Ramiro de la Vega
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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. 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?

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.

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. 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?

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Ramiro de la Vega
Ramiro de la Vega
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Does this property of a first-order structure imply categoricity?

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.