Is there an infinite singular cardinal $\kappa>0$ such that there is a set $E\subseteq{\cal P}(\kappa)$ with the following properties?

1. $|e| < \kappa$ for all $e\in E$,
2. whenever $\alpha\neq\beta\in \kappa$ there is $e\in E$ with $\{\alpha,\beta\} \subseteq e$, and
3. if $e_1\neq e_2\in E$ then $|e_1\cap e_2| = 1$.

(There can be no infinite regular cardinal with this property.)

Is there an infinite singular cardinal $\kappa$ such that there is a set $E\subseteq{\cal P}(\kappa)$ with the following properties?

1. $|e| < \kappa$ for all $e\in E$,
2. whenever $\alpha\neq\beta\in \kappa$ there is $e\in E$ with $\{\alpha,\beta\} \subseteq e$, and
3. if $e_1\neq e_2\in E$ then $|e_1\cap e_2| = 1$.

(There can be no infinite regular cardinal with this property.)