Uniqness of sets with certain property let $\kappa$ be a cardinal and let $A \subseteq  \{X \subseteq \kappa: |X| = 2 \}$ be a set with the propety:
"for each disjoint pair of sets $B,C \subseteq \kappa$ with cardinality < $\kappa$ there exists $\alpha \in \kappa - (B \cup C)  $ such that for each $\beta \in B , $ $\{\alpha,\beta\} \in A$ and
for each $\gamma \in C  $ , $\{\alpha,\gamma\} \notin A$"
now assume $A'$ also share the above propery, I need to prove the existence of an invertible function $f:\kappa \rightarrow \kappa$ such that for each $\alpha,\beta \in \kappa$ the following holds:
$\{\alpha,\beta\} \in A $ iff $\{f(\alpha),f(\beta)\} \in A' $
I really have no idea where to start, any help?
 A: Define $f$ inductively on $\kappa$.  Suppose that $f \restriction \beta$ has been appropriately defined.  Set 
$$\begin{gather}
B = \{ \alpha < \beta : \{ \alpha , \beta \} \in A \} \\
C = \{ \alpha < \beta : \{ \alpha , \beta \} \notin A \}.
\end{gather}$$
Now there is a $\delta\in \kappa \setminus ( f [ B ] \cup f [ C ] )$ such that $$\begin{gather}
( \forall \alpha \in B ) ( \{ f(\alpha) , \delta \} \in A^\prime ) \\
( \forall \alpha \in C ) ( \{ f(\alpha) , \delta \} \notin A^\prime ) \\
\end{gather}$$
so we set $f(\beta) = \delta$.

If you instead require that $f$ is bijective, then you proceed mostly as above, but instead inductively construct a sequence $\langle f_\beta \rangle_{\beta < \kappa}$ of functions such that 


*

*$f_\beta$ is injective;

*$\beta \subseteq \mathrm{dom} ( f_\beta )$;

*$\beta \subseteq \mathrm{ran} ( f_\beta )$;

*$\{ \alpha , \gamma \} \in A$ iff $\{ f(\alpha) , f(\gamma) \} \in A^\prime$ for all $\alpha , \gamma \in \mathrm{dom} ( f_\beta )$; and

*$f_\beta \supseteq f_\delta$ for $\delta \leq \beta$.


(This should be reminiscent of the back-and-forth argument to show that all dense linear orders without endpoints are order isomorphic.)
At limit steps we just take unions.
If $f_\beta$ is appropriately defined we first define $f_\beta^\prime$ to extend $f_\beta$ so that $\beta \in \mathrm{dom} (f_\beta^\prime)$ and still satisfies (1) and (4) above.  Next we extend $f_\beta^\prime$ to $f_{\beta + 1}$ so that $\beta \in \mathrm{ran} (f_{\beta+1})$  and also satisfies (1) and (4).  The details for these extensions are very similar to what I have outlined above, and so I will omit them.
