Let $\kappa$ be an infinite cardinal. Then we call $F\subseteq{\cal P}(\kappa)$ a *[Fano](https://en.wikipedia.org/wiki/Fano_plane) 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 $\kappa$?



Infinite Fano planes
Let $K$ be a field of cardinality $\kappa$. Fix a bijection between $\kappa$ and the projective plane over $K$. Then let $F$ be (the image under this bijection of) the set of lines in the projective plane.


