= Solution
The <Delta-system lemma> states that every uncountable family of finite <sets> contains an uncountable subfamily $\mathcal D$ and a fixed finite root $r$ such that
$$
\boxed{A\cap B=r\quad\text{for all distinct }A,B\in\mathcal D.}
$$
At a regular uncountable <cardinal> $\kappa$, a family of $\kappa$ finite <sets> has a <Delta-system> subfamily of size $\kappa$. The root may be empty. Neither version asserts that all <sets> in the original family have the same intersection.
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