Let $\kappa$ be an infinite cardinal. Then we call $F\subseteq{\cal P}(\kappa)$ a Fano plane on $\kappa$ if

1. $\bigcup F = \kappa$, $|F|=\kappa$, and $a\in F \implies |a| = \kappa$,
2. if $a\neq b\in F$ then $|a\cap b|=1$, and
3. if $x\neq y\in\kappa$ there is $a\in F$ such that $x,y\in a$ (note that by 2. above, this $a$ is uniquely determined).

Is there a Fano plane on every infinite cardinal $\kappa$?

Let $\kappa$ be an infinite cardinal. Then we call $F\subseteq{\cal P}(\kappa)$ a Fano plane on $\kappa$ if

1. $\bigcup F = \kappa$; $|F|=\kappa$; and $|a| = \kappa$ for all $|a|\in F$,
2. if $a\neq b\in F$ then $|a\cap b|=1$, and
3. if $x\neq y\in\kappa$ there is $a\in F$ such that $x,y\in a$ (note that by 2. above, this $a$ is uniquely determined).

Is there a Fano plane on every infinite cardinal $\kappa$?