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I wonder if there is an English translation of Fraïssé's paper "Sur l’extension aux relations de quelques propriet es desordres", appeared in Annales Scientifiques de l'Ecole Normale Superieure. Troisieme Śerie 71 (1954), 363–388." If not, where can I find the french paper (pdf)?

Also I'd like to know which text on model theory discuss details on Fraïssé theory, with one main result being a countable homogeneous structure is completely determined by its age. (The age of a countable structure is the collection of all its finitely generated substructures.)

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I'm not sure about an English translation of that particular paper, but Fraïssé's book Theory of Relations is available in English translation (this includes what is today called Fraïssé theory, in Section 11.1).

I agree with HallaSurvivor's comment that Hodges' A Shorter Model Theory is the best textbook reference for Fraïssé theory.

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  • $\begingroup$ @ Alex Kruckman, thanks. One more question, current model theory generally consider Fraïssé limit instead of direct limit because Fraïssé limit is more general than direct limit, right? $\endgroup$ – hermes May 25 at 3:51
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    $\begingroup$ @hermes More general? No, just different. Direct limit is a construction you do to a chain of structures. Fraïssé limit is a construction you do to a Fraïssé class of finite structures (which are not canonically arranged in a chain). The direct limit produces the "smallest" structure containing the chain (in the sense of its universal property), while the Fraïssé limit produces the "richest" countable structure containing all the structures in the class (in the sense of homogeneity). $\endgroup$ – Alex Kruckman May 25 at 14:09
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    $\begingroup$ The direct limit construction occurs all the time in model theory. One place it appears is in the construction of Fraïssé limits: the proof goes by cleverly constructing a chain of structures from the class (which witnesses all possible extensions in a precise sense) and then taking the direct limit of the chain. $\endgroup$ – Alex Kruckman May 25 at 14:14
  • $\begingroup$ Great. Which book contain the proof of the construction of Fraïssé limits by constructing a chain of structures from the class and then taking the direct limit of the chain? $\endgroup$ – hermes May 25 at 14:36
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    $\begingroup$ @hermes Literally any book which covers Fraïssé limits. It's the standard proof (though many authors will write "union" instead of "direct limit", because the direct limit along a chain of inclusion maps is just the union). Again, Hodges is a good choice. $\endgroup$ – Alex Kruckman May 25 at 15:10

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