# Are strongly complete regular linear hypergraphs on $\omega$ isomorphic?

This is a related question to an older one.

If $$H_i = (V_i, E_i)$$ are hypergraphs 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 strongly complete regular linear hypergraph on $$\omega$$ if

1. $$|e| = \aleph_0$$ for all $$e\in E$$,
2. $$e_1\neq e_2\in E \implies |e_1\cap e_2| = 1$$, and
3. for all $$m,n\in\omega$$ there is $$e\in E$$ such that $$\{m,n\}\subseteq e$$.

Question. If $$H_i = (\omega, E_i)$$ are strongly complete regular linear hypergraphs for $$i = 1,2$$, are $$H_1$$ and $$H_2$$ necessarily isomorphic?

If $$H_1= P^2(\mathbb{Q})$$ (the projective plane defined on $$\mathbb{Q}$$ with the points as vertices and lines as hyper-edges), and $$H_2$$ the Moulton plane defined on $$\mathbb{Q}$$, then $$H_1$$ and $$H_2$$ are both strongly complete regular linear hypergraphs but they are not isomorphic.

The reason is that, if $$H_1$$ and $$H_2$$ are isomorphic, Desargues' theorem would hold on both of them or neither of them, but Desargues' theorem holds on $$H_1$$ and not $$H_2$$.