Im having trouble solving this problem:
Let $M$ be a saturated structure of cardinality $\kappa$. Let $A\subseteq |M|$ with $|A|<\kappa$. Then there is a type $p\in S_1(A)$ with two different realizations.
I have tried to assume towards contradiction that all types in $S_1(A)$ have only one realization. and from there to conclude that for any $a\in|M|$ we get that $tp(a/A)$ are all the possible types there are in $S_1(A)$ and from there to isolate enough types to get a contradiction to the compactness of $S_1(A)$ as a topological space. but i didn't manage to continue to show this.
I will be happy to get some help thanks
Start by showing that for any type $p(x)\in S_1(A)$, if $p$ has exactly one realization in $M$, then there is some formula $\varphi(x)\in p$ such that $T\models \exists ! x\, \varphi(x)$. Then use compactness to show that there's a type which avoids all such formulas $\varphi$.
Your idea involving isolated types will also work ($\varphi$ isolates $p$ above), but you would have to break into cases, since it's possible that every type over $A$ is isolated. (But only if $A$ is finite and $T$ is $\aleph_0$-categorical! Then use the pigeonhole principal to find that one of the finitely many types over $A$ is realized many times.)