If $H\_i = (V\_i, E\_i)$ are [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) for $i=1,2$ then we say they are *isomorphic* if there is a bijection $f: V\_1 \to V\_2$ such that for $A \subseteq V\_1$ we have $$A\in E\_1 \text{ if and only if } f(A) \in E\_2.$$
We say that $H=(\omega, E)$ is a *complete regular linear* hypergraph on $\omega$ if 


1. $e\_1\neq e\_2\in E \implies |e\_1\cap e\_2| = 1$ and
2. for all $n\in \omega$ we have $|\{e\in E: n \in e\}| = \aleph\_0$.


**Question.** Assuming ${\sf ZFC}$, if $H\_i = (\omega, E\_i)$ are complete regular linear hypergraphs for $i = 1,2$, are $H\_1$ and $H\_2$ necessarily isomorphic?



Are complete regular linear hypergraphs on $\omega$ isomorphic?
If $K$ is a field of cardinality $\aleph\_0$, then the points and lines of the projective plane over $F$ constitute a complete regular linear hypergraph. The field $K$ can be recovered (up to isomorphism) from the hypergraph, so this produces lots of non-isomorphic such hypergraphs.


