Let $H=(V,E)$ be a hypergraph such that $|V|$ is infinite, and the following statements hold:

- if $a\neq b\in E$ then $|a\cap b|\leq 1$, and
- every vertex $v\in V$ is contained in at least $2$ members of $E$.

Is there $P\subseteq E$ such that the members of $P$ are pairwise disjoint, and $\bigcup P = V$? (This would amount to a kind of matching in hypergraphs.)

finitesets, do they? In the first example I thought of, the vertices are the points of the real projective plane, the points are the lines; two edges intersect in exactly one vertex, each vertex belongs to continuum many edges. $\endgroup$ – bof Dec 1 '17 at 9:27