Let $A$ be an infinite set. Is it always possible to find a subset $\mathscr C$ of the power set $\mathscr P(A)$ of $A$, which has the same cardinality with $\mathscr P(A)$, and has the following property: If $B,C\in\mathscr C$ and $B\ne C$, then the cardinality of $B\cap C$ is smaller than the cardinality of $A$.
Such subset can be found if $A=\mathbb N$. What if $A$ is uncountable?
I have asked this question on MathOverflow in the past. You can find the question here, and as Andreas Blass points out in his answer, it is consistent that $|A|=\aleph_1$ and there are no $2^{\aleph_1}$ subsets of $A$ that every two distinct sets meet on a countable set.