Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\subseteq\mathscr K$ with $\bigcap\,\mathscr A\not=\emptyset$ ?

Assuming the continuum hypothesis, the answer is negative. First of all, we can index ${\cal K}$ by $[0,1]$, let $K_x$ be the set in ${\cal K}$ with index $x$. Now wellorder $[0,1]$ in a way corresponding to the first uncountable ordinal. For any $x$, there are only countably many points preceding $x$ in the well ordering. We can therefore choose $K_x$ so that it does not contain any point that precedes $x$. Now let $A$ be any uncountable set and consider $\bigcap_{x\in A} K_x$. For any $y$, there is an $x\in A$ which comes after $y$. Hence this set must be empty. 


Assuming $MA_{\aleph_1}$, the answer is positive. Let $P$ be the collection of all positive measure finite intersections of elements of $\mathcal{K}$, ordered by inclusion. Then $P$ is a ccc uncountable partial order so (by $MA_{\aleph_1}$) it contains an uncountable centered subset $Q \subseteq P$. If we let $\mathcal{A}$ be the collection of all elements of $\mathcal{K}$ that contain some element of $Q$, it follows that $\mathcal{A}$ has the finite intersection property and therefore $\bigcap \mathcal{A} \neq \emptyset$. Edit: A subset $Q$ of a poset $P$ is called centered if any finite $F \subseteq Q$ has a lower bound in $P$. It was proved by Velickovic and Todorcevic in "Martin's axiom an partitions" (1987) that $MA_{\aleph_1}$ is equivalent to the statement that every ccc uncountable partial order contains an uncountable centered subset. 

