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In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ $fragment$ from an infinitary lenguage, and a complete theory $T$ ).

He starts proving that $D(F,T)$ (the set of all possible $F$-diagrams of models of $T$) is Borel and then he constructed a continous map $\psi$ such that $S_n(F,T)$ is the image of $D(F,T)$ under this map.

Does anyone know what progress has been made about the properties we can obtain from models of a non scattered theory using that $D(F,T)$ is Borel? ( aside from the fact that $|S_n(F,T)|$ is $\aleph_0$ or $2^{\aleph_0}$ ).

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