Suppose that $\Gamma$ is an infinite set. Let us say that a family $\mathscr A$ of subsets of $\Gamma$ is almost disjoint, whenever for any two distinct sets $A_1, A_2\in \mathscr{A}$ the intersection $A_1\cap A_2$ has cardinality strictly less than $|\Gamma|$ and for any $A\in \mathscr{A}$ we have $|A|=|\Gamma|$.
Does there always exist an almost disjoint family of cardinality $2^{|\Gamma|}$, or at least bigger than $|\Gamma|$?
When $\Gamma$ is countable, then of course this is the case, but I have a feeling this should fail for singular cardinals.
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