TY - JOUR

T1 - Partial λ-geometries of small nexus

AU - Cameron, Peter J.

AU - Drake, David A.

PY - 1980/1/1

Y1 - 1980/1/1

N2 - In a partial λ-geometry, each two points are joined by 0 or λ blocks; each two blocks have 0 or λ points in common; the block size k is constant; and for each nonflag (p, G), there are precisely e blocks X with p in X such that X∩G is not empty. Generalized quadrangles are partial 1–geometries with nexus e = 1. If λ = 2, then e ⩾ 3; and the first author has determined that partial 2–geometries with nexus 3 exist precisely for the values k = 3, 4, 8, 24. We prove (1) that if λ > 2 and k > e + 1, then e > 2λ (2) that λ = 3, e = 7 implies k is one of 7, 15, 21, 24 or 36. We call a partial λ-geometry extremal if |G∩H∩K| > 1 implies |G∩H∩K| ⩾ λ. There are no extremal partial λ-geometries with e = λ2- λ. Such a geometry with e = λ2- λ + 1 and k > e is called a λ-quadrangle. We determine all λ-quadrangles with λ > 2. They are constructed from quadratic forms of Witt index 4 on finite 8–dimensional vector spaces.

AB - In a partial λ-geometry, each two points are joined by 0 or λ blocks; each two blocks have 0 or λ points in common; the block size k is constant; and for each nonflag (p, G), there are precisely e blocks X with p in X such that X∩G is not empty. Generalized quadrangles are partial 1–geometries with nexus e = 1. If λ = 2, then e ⩾ 3; and the first author has determined that partial 2–geometries with nexus 3 exist precisely for the values k = 3, 4, 8, 24. We prove (1) that if λ > 2 and k > e + 1, then e > 2λ (2) that λ = 3, e = 7 implies k is one of 7, 15, 21, 24 or 36. We call a partial λ-geometry extremal if |G∩H∩K| > 1 implies |G∩H∩K| ⩾ λ. There are no extremal partial λ-geometries with e = λ2- λ. Such a geometry with e = λ2- λ + 1 and k > e is called a λ-quadrangle. We determine all λ-quadrangles with λ > 2. They are constructed from quadratic forms of Witt index 4 on finite 8–dimensional vector spaces.

UR - http://www.scopus.com/inward/record.url?scp=0011434093&partnerID=8YFLogxK

U2 - 10.1016/S0167-5060(08)70692-7

DO - 10.1016/S0167-5060(08)70692-7

M3 - Article

AN - SCOPUS:0011434093

SN - 0167-5060

VL - 6

SP - 19

EP - 29

JO - Annals of Discrete Mathematics

JF - Annals of Discrete Mathematics

IS - C

ER -