Line-closed subsets of Steiner triple systems and classical linear spaces

Camina, A.R. and Miller, A.A. (1996) Line-closed subsets of Steiner triple systems and classical linear spaces. Journal of Statistical Planning and Inference, 56(1), pp. 65-77. (doi: 10.1016/S0378-3758(96)00010-9)

Abstract

A proper non-empty subset C of the points of a linear space = (P,L) is called line-closed if any two intersecting lines of , each meeting C at least twice, have their intersection in C. We show that when every line has k points and every point lies on r lines the maximum size for such sets is r + k − 2. In addition it is shown that this cannot happen for projective spaces PG(n,q) unless q = 2, nor can it be obtained for affine spaces AG(n,q) unless n = 2 and q = 3. However, for all odd values of r there exist Steiner triple systems having such maximum line-closed subsets.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Miller, Professor Alice
• Authors: Camina, A.R., and Miller, A.A.
• College/School: College of Science and Engineering > School of Computing Science
• Journal Name: Journal of Statistical Planning and Inference
• ISSN: 0378-3758
• ISSN (Online): 1873-1171