Is there an infinite cardinal $\kappa$ with a collection of subsets ${\cal E}$ of $\kappa$ with the following properties?


1. $\bigcup {\cal E} = \kappa$,
2. $e \neq f \in {\cal E}$ implies $|e \cap f|=1$
3. $|e|<\kappa$ for all $e\in {\cal E}$, and
4. not all members of ${\cal E}$ have the same cardinality.



Infinite complete linear hypergraphs with edges of different sizes
Partition $\omega \setminus \{0\}$ into infinitely many finite sets $I\_n$ such that $\vert I\_n \vert=n$. Define $\mathcal{E}:=\{I\_n \cup \{0\} \colon n \in \omega\}$.


