Pentagonal geometries with block sizes 3, 4, and 5

Forbes, Anthony D. (2021). Pentagonal geometries with block sizes 3, 4, and 5. Journal of Combinatorial Designs, 29(5) pp. 307–330.

DOI: https://doi.org/10.1002/jcd.21768

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 2³⁵, 2⁷¹, and 10²³) we significantly extend the known existence spectrum for PENT (5, r) .

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