The well-known $\Delta$-system lemma states:
Each uncountable family of finite sets has an uncountable sub-family with a root. Precisely:
$$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$ $$(\exists \Delta~\exists F'\subseteq F~~~|F'|>\aleph_0~\wedge~\forall X,Y\in F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$
For which uncountable families of finite sets like $F$ is the following version of $\Delta$-system lemma true?
$$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in F~~~|X|<\aleph_0)\longrightarrow$$ $$(\exists \Delta~\exists F'\subseteq F~~~|F'|=|F|~\wedge~\forall X,Y\in F'~(X\neq Y\rightarrow X\cap Y=\Delta))$$
I am looking for an "intuitive" necessary and sufficient condition on $F$.
Let $\kappa$ be a singular cardinal with $\text{cf}(\kappa) = \lambda < \kappa$.
Write $\kappa - \lambda = \bigsqcup_{\alpha < \lambda} U_\alpha$ where $|U_\alpha|<\kappa$.
Let $F = \{ \ \{\alpha, \beta\} : \alpha < \lambda \wedge \beta \in U_\alpha\}$.
Note $|F| = \kappa$. Since the $U_\alpha$ are disjoint partition of $\kappa - \lambda$, the root of any $\Delta$-system must be a singleton $\{\alpha\}$ for some $\alpha < \lambda$. However, if $x \cap y = \{\alpha\}$ for $x,y \in F$, then $x,y \in U_\alpha$. $|U_\alpha| < \kappa$. Hence there is no $\Delta$-system of size $\kappa$.