Does there exist a finite hyperbolic geometry in which every line contains at least 3 points, but not every line contains the same number of points?

Question

It seems to me that the answer should be yes, but my naive attempts to come up with an example have failed.

Just to clarify, by finite hyperbolic geometry I mean a finite set of points and lines such that

1. every pair of points determines determines a unique line
2. every line contains at least two points
3. there are at least two distinct lines
4. for every line L and every point p not on the line, there are at least two distinct lines which are incident with p which do not intersect L

It is easy enough to show that in finite euclidean geometries (i.e. affine planes) all lines have the same number of points, and in finite elliptic geometries in which every line contains at least $3$ points (i.e. projective planes) all lines have the same number of points.

It is also easy enough to come up with an example of a finite hyperbolic geometry in which not every line has the same number of points (but some lines have exactly two points). e.g. Let the points be $\{1,2,3,4,5,6\}$ and let the lines be the set $\{1,2,3\}$, together with all of the $2$-element subsets except for $\{1,2\}$, $\{1,3\}$, $\{2,3\}$.

Answer 1

Take a 2-$(v,4,1)$ design on $v$ points and delete one block, along with the four points on it. In the original system each point is on exactly $(v-1)/3$ blocks, so if we assume $v\ge25$ the geometry we get by deleting one block is hyperbolic, all blocks have at least three points, and there are blocks of size three and size four.

Answer 2

Let each line of a projective plane $\ P\ $ have $\ p\ge 5\ $ points. Let lines $\ L_0\,\ L_1\,\ L_2\ $ of $\ P\ $ have empty intersection $\ L_0\cap L_1\cap L_2\ =\ \emptyset.\ $ Define

$$ H\ :=\ P\setminus(L_0\cup L_1\cup L_2) $$

The H-lines are defined as sets $\ H\cap L,\ $ where $\ L\ $ is a projective line in $\ P.$

Then $\ H,\ $ together with H-lines, is a hyperbolic plane such that each H-line has $\ p-2\ $ or $\ p-3\ $ points, where both these cardinalities do happen.

Let me present a proof of property 4: INDEED, let $\ x\in H\setminus L,\ $ where L is an H-line, and let $\ L'\ $ be the projective extension of $\ L.\ $ For each $\ y\in L'\cap M',\ $ where $\ M'\ $ is one of the removed projective lines, we get the H-line $\ xy\cap H\ $ which is parallel to $\ L,\ $ where $\ xy\ $ is the projective line passing through $\ x\ $ and $\ y.$

There is also a removed line $\ N'\ $ which doesn't pass through $\ y\ $ (the intersection of all removed lines is assumed to be empty). Thus $\ \exists_z\ y\ne z\in L'\cap N',\ $ and $\ xz\cap H\ $ is another H-line parallel to $\ L\ $ different from the previous parallel line.

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More generally, one may consider any family of three or more projective lines (instead of three of them), which have an empty intersection. Then, with a bit of care, one gets (infinitely) more required examples with more than two different cardinalities of the H-lines.

Answer 3

You can find out more about different approaches to constructing finite hyperbolic planes from this paper and consulting the references in this paper:

https://projecteuclid.org/download/pdf_1/euclid.mmj/1028999550

I believe there are still some open questions about the existence of such planes where there are at least 3 points on a line, lines have equal numbers of points, and there is the same number p of parallels to each line l through a point P not on l.