On pentagonal geometries with block size 3, 4 or 5

Abstract

Let PLS(k, r) be a partial linear space which is both uniform, i.e. every line has the same cardinality k ≥ 2, and regular, i.e. every point is incident with the same number r ≥ 1 of lines. In a recent paper (J. Combin. Des. 21 (2013), 163–179), Ball, Bamberg, Devillers & Stokes introduced the concept of a pentagonal geometry PENT(k, r) as a PLS(k, r) in which all the points not collinear with any given point are themselves collinear. They also determined the existence spectrum for k = 1 or 2 and r = k or k+1. In this paper we prove that the existence spectrum for PENT(3, r) is r ≡ 0 or 1 (mod 3) except r = 4 or 6.We also prove that there exists a PENT(4, r) for r ≡ 1 (mod 8) and a PENT(5, r) for r ≡ 1 (mod 5), r ≠6, apart from nine possible exceptions. Further we construct an infinite class of pentagonal geometries PENT(2m, 2m+1 + 1), m ≥ 1, and a PENT(6, 13).