Proving/Disproving there are always two uncountable sets whose intersection is uncountable. I have been trying to prove the following:
Let $\mathcal{C}$ be an uncountable family of uncountable subsets of $\mathbb{R}$. Either prove or disprove that there are always two sets in $\mathcal{C}$ whose intersection is an uncountable set.
My intuition tells me that the statement is true and that it is connected to the axiom of choice. Although, no matter what I try, it doesn't seem to go anywhere.

The statement is false as noted by @Hanul Jeon in comments.
Consider the following uncountable collection of uncountable disjoint subsets of $\mathbb{R}^2$:
$$\mathcal{U}=\big\{\{x\}\times\mathbb{R}\ \big|\ x\in\mathbb{R}\big\}$$
Then consider any bijection $f:\mathbb{R}^2\to\mathbb{R}$ and note that
$$f(\mathcal{U})=\big\{f(U)\ \big|\ U\in\mathcal{U}\big\}$$
is an uncountable collection of pairwise disjoint uncountable subsets of $\mathbb{R}$.
