Pentagonal geometries with block sizes 3, 4, and 5

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 1 0 23 ) we significantly extend the known existence spectrum for PENT ( 5 , r ) .