SCHWEITZER 1976 Problem 2

Let $G$ be an infinite graph such that for any countably infinite vertex set $A$ there is a vertex $p$ , not in $A$ , joined to infinitely many elements of $A$ . Show that $G$ has a countably infinite vertex set $A$ such that $G$ contains uncountably infinitely many vertices $p$ joined to infinitely many elements of $A$ .

P. Erdos, A. Hajnal