We say that a hypergraph $(\mathbb{N}, E)$ where $E\subseteq {\cal P}(\mathbb N)$ is *perfectly dense* if 


1. $\mathbb{N}\notin E$,
2. all $e\in E$ are infinite,
3. $e\_1, e\_2 \in E$ implies $|e\_1\cap e\_2| = 1$, and
4. for all $m\neq n\in \mathbb{N}$ there is $e\in E$ such that $\{m,n\}\subseteq e$.


If $(\mathbb{N}, E\_i)$ are perfectly dense for $i=1,2$, are these hypergraphs necessarily [isomorphic](https://mathoverflow.net/questions/267355/non-isomorphic-hypergraphs-on-omega)? If not, how large can a collection of perfectly dense, pairwise non-isomorphic hypergraphs be?


(Will also accept posts answering the first question only, but I am curious about the second, too.)



Are two "perfectly dense" hypergraphs on $\mathbb{N}$ necessarily isomorphic?
There are continuum-many pairwise non-isomorphic perfectly dense hypergraphs. Below is a sketch of a proof.


Given a countably infinite field $\mathbb{K}$, the projective plane $\mathbb{KP}\_2$ over $\mathbb{K}$ can be seen as a perfectly dense hypergraph, where vertices are points and edges are lines. I will show that from the hypergraph structure of $\mathbb{KP}\_2$, it is possible to recover the field $\mathbb{K}$ up to isomorphism. Since there are $2^{\aleph\_0}$ pairwise non-isomorphic countably infinite fields (take for instance $\mathbb{K}=\mathbb{Q}\left(\{\sqrt{p} \mid p \in A\}\right)$ for $A$ a subset of prime numbers), this will be enough to conclude.


So let $\mathbb{K}$ be a contably infinite field and $P$ be a projective plane over $\mathbb{K}$. I choose three arbitrary and pairwise distinct lines $L\_\infty$ (the *line at infinity*), $L\_x$ (the *$x$-axis*) and $L\_y$ (the *$y$-axis*) in $P$. Let $P'=P \setminus L\_\infty$, $L'\_x = L\_x \cap P$ and $L'\_y = L\_y \cap P$. Let $0\_P$ be the intersection between $L\_x$ and $L\_y$, and choose arbitrarily $1\_x \in L'\_x \setminus \{0\_P\}$ and $1\_y \in L'\_y \setminus \{0\_P\}$. Since $P'$ is an affine plane, there exists a unique affine isomorphism $\varphi \colon \mathbb{K}^2 \to P'$ with $\varphi(0, 0) = 0\_P$, $\varphi(1, 0) = 1\_x$ and $\varphi(0, 1) = 1\_y$. We can transfer the field structure of $\mathbb{K}$ to $L'\_x$ by the bijection $a \mapsto \varphi(a, 0)$; we still denote by $+$ and $\cdot$ the field operations on $L'\_x$. We show that these operations are actually definable only from $L\_x$, $L\_y$, $L\_\infty$, $1\_x$, $1\_y$ and from the hypergraph structure of $P$; since $L\_x$, $L\_y$, $L\_\infty$, $1\_x$, $1\_y$ have been chosen arbitrarily, it will be enough to conclude.


A few more notation. For $A, B \in P$, I will denote by $(AB)$ the line passing through $A$ and $B$. Say that two lines in $P$, distinct from $L\_\infty$, are *parallel* if they are equal or are distinct and intersect on $L\_\infty$, and that a line $L$ in $P$ is *horizontal* (resp. *vertical*) if it is distinct from $L\_\infty$ and parallel to $L\_x$ (resp. $L\_y$). The notions of parallelism, horizontality and verticality are definable from $L\_x$, $L\_y$, $L\_\infty$ and from the hypergraph structure on $P$.


Now let $A, B \in L'\_x$; we show how to define $A + B$ and $A\cdot B$. Let $A'$ be the intersection of $L\_y$ and of the parallel to $(1\_x1\_y)$ passing through $A$. Let $C$ be the intersection of the vertical line passing through $B$ and of the horizontal line passing through $A'$. Then $A+B$ is the intersection of $L\_x$ and of the parallel to $(1\_x1\_y)$ passing through $C$, and $A \cdot B$ is the intersection between $L\_x$ and the parallel to $(1\_yB)$ passing through $A'$.


P.S.: I would be curious to see a simpler purely combinatorial proof; I am sure such a proof should exist.


