Together with [@TarasBanakh][1] we faced the problem described in the title. Let me start with definitions.

A *linear space* is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of subsets of $S$ where for any distinct points $x,y\in S$ there exists a unique line $L\in\mathcal L$ containing $x$ and $y$. We'll denote this line as $\overline{xy}$.

A *hull* of a set $A$ we will call a smallest set $\overline{A}$ such that $A \subset \overline{A} \land \forall x,y \in \overline{A} (x \not= y \rightarrow \overline{xy} \subset \overline{A}$). Having this hull definition we can easily define dimensional-based structures. In order to simplify writing, we will just write $\overline{a_1a_2...a_n}$ instead of $\overline{\{a_1, a_2, ..., a_n\}}$.

We'll call a set $A$ *collinear* if $\forall x, y \in A(x\not= y \rightarrow \overline{A} = \overline{xy})$. Set $P$ is called a *plane* if exist 3 non-collinear points $x, y, z\in S:P=\overline{xyz}$. Similarly we can define set *planar* if hull of any non-collinear 3-subset of it is the same plane $P$. At last, we'll call a set $S$ *3-dimensional space* if there exist 4 distinct non-collinear and non-planar points $x,y,z,w \in S:S=\overline{xyzw}$.

We'll call 3-dimensional space *balanced* (due to BIBD) when cardinality of all lines is equal and cardinality of all planes is equal as well.

And at last, we'll call a 3-dimensional space $S$ *Lobachevsky* if for every plane $P\subset S$, line $L \subset P$ and point $x \in P \setminus L$ there exist at least two distinct lines $\Lambda_1, \Lambda_2$ in $S$ such that $x\in \Lambda_i\subset P\setminus L$. At last we can formulate the problem.

>**Problem.** Does there exist *finite balanced 3-dimensional Lobachevsky* space $S$?

Some comments. Obviously there exist infinite any-dimensional [Lobachevsky space][3]. Finite [affine][4] and [projective spaces][5] are also well-known. Finite balanced Lobachevsky planes also [exist (simplest example)][6]. If we need Lobachevsky plane where there exist no [Pasch configuration][7], then we have [unitals][8]. So, we can see that there are infinitely-many balanced Lobachevsky planes. Still, I was not able to find any example of Lobachevsky 3-dimensional space.

Main difficulty is that such spaces should have lots of points. General formula for point number is very probably the following: $|S|=1 + (|L|-1)(1 + (k + |L| - 1) + (k + |L| - 1)^2)$ where $|L|\geq 3$ and $k\geq 2$ are number of points in line and number of "parallel" lines respectively (still could be wrong). Also not all $k$ are suitable. For instance, for 3-point lines only $k \equiv 0, 1 (\mod 3)$ are possible. According to this calculations, in case of 3-point lines, smallest example should have $63 = 1 + 2 * (1 + 5 + 25)$ points, next should have $87=1 + 2 * (1 + 6 + 36)$. If we're talking about 4-point lines, smallest should have $172 = 1 + 3 * (1 + 7 + 49)$, next $220 = 1 + 3 * (1 + 8 + 64)$ points. Therefore finding it by generative methods is very complicated (for instance, I checked all non-isomorphic $BIBD(63,3,1)$ generated by difference families and lots of similar $BIBD(87,3,1)$). Despite Lobachevsky 13-point and 15-point planes are [difference-family-based][9], there is no difference-family-based $BIBD(63,3,1)$ and very probably no difference-family-based $BIBD(87,3,1)$.

I also tried to generate 3-dimensional analogues to unitals, Denniston maximal arcs, derived Steiner Triple Systems from Steiner Quadruple Systems and so on. I spent like 3 months of searching by computer and in literature/articles without any success. This is the motivation for problem above.


  [1]: https://mathoverflow.net/users/61536/taras-banakh
  [2]: https://en.wikipedia.org/wiki/Steiner_system
  [3]: https://en.wikipedia.org/wiki/Hyperbolic_space
  [4]: https://en.wikipedia.org/wiki/Vector_space
  [5]: https://en.wikipedia.org/wiki/PG(3,2)
  [6]: https://math.stackexchange.com/a/104110
  [7]: https://mathworld.wolfram.com/PaschConfiguration.html
  [8]: https://mathoverflow.net/a/457894/515881
  [9]: https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/designs/bibd.html#sage.combinat.designs.bibd.BIBD_from_difference_family