Pentagonal geometries with block sizes 3, 4 and 5

Abstract

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line, or block, is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. An opposite line pair in a pentagonal geometry consists of two parallel lines such that each point on one of the lines is not collinear with precisely those points on the other line. We give a direct construction for an infinite sequence of pentagonal geometries with block size 3 and connected deficiency graphs. Also we present 39 new pentagonal geometries with block size 4 and five with block size 5, all with connected deficiency graphs. Consequentially we determine the existence spectrum up to a few possible exceptions for PENT(4, $r$) that do not contain opposite line pairs and for PENT(4, $r$) with one opposite line pair. More generally, given $j$ we show that there exists a PENT(4, $r$) with $j$ opposite line pairs for all sufficiently large admissible $r$. Using some new group divisible designs with block size 5 (including types $2^{35}$, $2^{71}$ and $10^{23}$) we significantly extend the known existence spectrum for PENT(5, $r$).