 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 thatAt 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.