On the existence of a certain graph/hypergraph pair

Question

Let $V$ be a finite set, $G$ a simple graph with vertex set $V$, and $H$ a hypergraph (i.e., set of subsets) with vertex set $V$ satisfying the following three conditions:

• each pair of elements of $V$ is contained in a unique hyperedge of $H$;

• for $x, y, z \in V$, if $x$ is $G$-adjacent to $z$ and $y$ is $G$-adjacent to $z$, then $z$ is also $G$-adjacent to every other element of the hyperedge containing $x$ and $y$;

• the restriction of $G$ to any hyperedge of $H$ is a perfect or near-perfect matching (i.e., a disjoint union of edges together with at most one isolated vertex).

Question: is it possible that there is $x\in V$ such that $x$ belongs to two (or more) hyperedges in $H$ of size greater than or equal to three?

Note: if we allow $V$ infinite, then this is possible. Indeed, we can take $V$ to be the set of all lines in $\mathbb{R}^n$ for $n\geq 3$, with two elements $G$-adjacent if they are orthogonal, and where the hyperedges of $H$ are all the two-dimensional subspaces.

Answer 1

I'm sorry, but I do not understand why you cannot model the same example over a finite field.

Let, say, $V$ be the set of non-isotropic points in the projective space $P^3(\mathbb F_p)$ for a large $p$, i.e., the points $(x:y:z:t)$ with $x^2+y^2+z^2+t^2\neq0$. Again, two points are connected in $G$ if they are orthogonal (the non-isotropy condition avoids loops), and the hyperedges in $H$ are projective lines (restricted to $V$). All the conditions are clearly satisfied (in the third condition, each hyperedge induces even a perfect matching --- again with the use of non-isotropy condition). Moreover, almost al hyperedges of $H$ will be large, if $p$ is such.