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
Thank you for help. :)