Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-07-07T03:59:29.079Z Has data issue: false hasContentIssue false

Linear spaces with line range {n−1, n, n + 1} and at most n2 points

Published online by Cambridge University Press:  09 April 2009

Lynn Margaret Batten
Affiliation:
Département de Mathématique, Université Libre de Bruxelles, C. P. 216 —Campus Plaine, Boulevard du Triomph, B-1050 Bruxelles, Belgique
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterize all finite linear spaces with pn2 points where n ≥ 8 for pn2 − 1 and n ≥ 23 for p = n2−1, and the line range is {n−1, n, n+1}. All such linear spaces are shown to be embeddable in finite projective planes of order a function of n. We also describe the exceptional linear spaces arising from p < n2−1 and n ≥ 4.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1980

References

Batten, L. M. and Totten, J. (1979). ‘On a class of linear spaces with two consective line degrees’ Ars Combinatoria (to appear).Google Scholar
Bose, R. C. and Shrikhande, S. S. (1963), ‘Embedding the complement of an oval in a projective plane of even order’, Discrete Math. 6, 305312.CrossRefGoogle Scholar
McCarthy, D. and Vanstone, S. A. (1977), ‘Embedding (r.1)-designs in finite projective planes’, Discrete Math. 19, 6776.CrossRefGoogle Scholar
Mullin, R. C. and Vanstone, S. A. (1976), ‘A generalization of a theorem of Totten’, J. Austral. Math. Soc. Ser. A 22, 494500.CrossRefGoogle Scholar
Totten, J. (1976), ‘Embedding the complement of two lines in a finite projective plane’, J. Ausral. Math. Soc. Ser A 22, 2734.CrossRefGoogle Scholar
de Witte, P. (1976), ‘The exceptional case in a theorem of Bose and Shrikhande’, J. Austral. Math. Soc. Ser A 24, 6478.CrossRefGoogle Scholar
de Witte, P. (1979a), ‘On the embeddability of linear spaces in projective spaces of order n’, Trans. Amer. Math. Soc. (to appear)Google Scholar
de Witte, P. (1979b), ‘Finite linear spaces with two unequal line degrees’ (to be submitted).Google Scholar