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Find a prime model and a countable $\omega$-saturated model of $Th((\mathbb{Q},+,0))$.

Define a function from $\mathbb{N}$ to $\mathbb{N}$ such that, for each n, $Th((\mathbb{Q},<))$ has not more than $f(n)$ $n$-types.


I know for the first part that i have to find a model $\mathfrak{A}$ of $Th((\mathbb{Q},+,0))$ such that every other model embedds elementary in this model $\mathfrak{A}$ (here we can use Fraisse's Method). I think that $(\mathbb{Q},+,0)$ self i such a model but is this true? But what about a counatble-saturated model. I know the definition but it is very difficult to understand this definition. Can someone give a hint?

For the second part: i know that for each n holds: $(q_0,\cdots,q_{n-1})$ in $Th((\mathbb{Q},<))$ realized the same n-type as $(r_0,\cdots,r_{n-1})$ iff for all i$,j: $q_i iff $r_i (this is a equivalence relation with finitly many equivalence classes). The statement is clear, but how to define such a function $f$ with this information?

Thank you for help. :)

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