Show that the following are equivalent:
- $M$ realizes all $\lambda$-types of $M$ over $\emptyset$ for any $\lambda<\kappa$.
- For any $N\equiv M$ and any $A \subseteq N$ with $|A|<\kappa$ there is a partial elementary map $f : N \to M$ with $A \subseteq dom(f)$.
Moreover, if $\kappa > |L|+\aleph_0$ then 1. and 2. are equivalent to:
- $M$ is $\kappa$-universal.
Throughout, I write "type" for "type over $\emptyset$."
The key piece of intuition, I think, should be the following: a "type" is just a description, of an object (or family of objects) which could exist. When I have a type $p$ with $\theta$-many variables, I think of that type as describing a collection of $\theta$-many elements, and telling me how they interact with each other (what relations hold and don't hold, etc.). The two key properties are completeness (which doesn't matter here so much) and consistency (which does - consistency means that the type is realized in some model, that is, there is some model with a bunch of things which are described by the type). Caution: just because something's possible doesn't mean it happens - I can write a type $p$ and have a model where nothing corresponding to $p$ occurs.
For example, the type $\{x>1+1+ . . . +1$ ($n$ times)$: n\in\omega\}$ is not realized in $(\mathbb{N}, <, +)$, but there are models of the theory of $(\mathbb{N}, <, +)$ in which it is realized, that is, models with some element $a$ which is $>1$, $>1+1$, $>1+1+1$, etc.
OK fine that type's not complete but you know what I mean.
Bottom line: think of "realizing a type" as "having a bunch of elements which look a certain way."
So with that in mind, here's a hint for (1) $\rightarrow$ (2): Fix $N\equiv M$ and $A\subseteq N$ with $\vert A\vert=\lambda<\kappa$. Now, by (1) we know that any $\lambda$-type $p$ is realized in $M$ by some elements $\overline{m}_p$; can you think of a type $p$ so that $A$ "looks like" $\overline{m}_p$? HINT: this type should describe $A$ in some way . . .
For (2)$\implies$(1), suppose I have a $\lambda$-type $p$. Being a type means $p$ is realized in some model . . . but maybe not $M$. (Of course, it will be, but we can't assume that - that's what we're trying to show.) So suppose $N$ is a model in which $p$ is realized, by some set $A$. Using (2), we can "transfer" $A$ to some tuple $B$ in $M$; what does this tuple have to do with $p$?
After you've worked these out, handling (3) should be easy.