Question about graphs A random graph is a graph $\mathcal{V} = (V,R)$ which is non directed and has no loops and such that for every pair of finite disjoint subsets $A$ and $B$ of vertices satisfying that there exists a vertex $c \in V$ such that $(c, a) \in R$ for every $a \in A$ and $(c, b) \in R$ for every $b \in B$.
I need to prove that the point $c$ can be specifically chosen in $V \setminus (A \cup B)$ (notice that the set $V$ is infinite). I've been looking on the web and every site I have found about random graphs said that this is either the definition or trivial.
What I've tried to show that if $c \in A \cup B$ then we have a contradiction, but I cannot find the contradiction... Also, the proof by induction seems to be quite difficult. Also, I am supposed to prove this without probability theory ideas.
Any help would be appreciated. Thanks in advance.
 A: In my opinion, unless I'm missing a simpler argument, this is not trivial; although the argument is short, it does take a trick. The idea is to apply randomness twice, with the first application being "preparatory" and designed to force the second application to choose something outside $A\cup B$.
Suppose $\mathcal{G}$ is a random graph and $A,B$ are disjoint finite subsets of $\mathcal{G}$. Applying randomness to the pair $(A\cup B,\emptyset)$ we get a vertex $v$ connected to all vertices in $A$ and $B$; by the no-loops condition we must have $v\not\in A\cup B$. Now apply randomness again to $(A,B\cup\{v\})$. A vertex $w$ which is connected to everything in $A$ but nothing in $B\cup\{v\}$ cannot lie in $A\cup B$ since everything in $A\cup B$ is connected to $v$. Forgetting about $v$, such a $w$ is then a vertex in $\mathcal{G}$ connected to everything in $A$, not connected to anything in $B$, and not in $A\cup B$.

Note that the above relies on the no-loops condition. This may seem a bit odd; in fact, a different two-step argument shows that it isn't necessary. First, you can show that every random graph must be infinite (if you haven't done this yet it's a good exercise). Now let $X$ be a set of vertices of size $\vert A\cup B\vert$ which is disjoint from $A\cup B$. Since $\vert A\cup B\vert<\vert\mathcal{P}(A\cup B)\vert=2^{\vert A\cup B\vert}$, there is some $Y\subseteq X$ such that no element of $A\cup B$ is connected to everything in $Y$ and not connected to anything in $X\setminus Y$. Now apply randomness to the pair $(A\cup Y, B\cup (X\setminus Y))$.
