Small almost complete arcs
type: Item
instance of: scholarly article
title: Small almost complete arcs
zbMATH Open document ID: 1027.51012
DOI: 10.1016/S0012-365X(01)00412-5
published in: Discrete Mathematics
publication date: 2002-12-02T00:00:00Z
review text: A \(k\)-arc in a projective plane is a set of \(k\) points no three of which are collinear. An arc is complete if it is not properly contained in another arc. \textit{L. Lunelli} and \textit{M. Sce} [Ist. Lombardo Accad. Sci. Lett., Rend., Sci. Mat. Fis. Chim. Geol., Ser. A 98, 3-52 (1964; Zbl 0131.36802)] proved that if \(K\) is a complete \(k\)-arc in a projective plane of order \(q\), then \(k > \frac{1}{2}(3 + \sqrt{8q+1}) \approx \sqrt{2q}\). A \(k\)-arc is complete if and only if every point of the plane is contained in at least one secant of the arc. The author constructs a sequence of \(k\)-arcs contained in irreducible conics of desarguesian planes of order \(q\), for some odd prime powers \(q\), such that \(k \lesssim 13(\log_2 q^2) \sqrt{q}\) and such that, if \(q\) goes to infinity, the ratio of the number of points not on a secant of the \(k\)-arc to the total number of points in the plane, goes to 0.
reviewed by: Luc Teirlinck
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 1838905
zbMATH Keywords: projective plane
zbMATH Keywords: complete arc
zbMATH Keywords: Baer subplane
author: Emanuela Ughi
MaRDI profile type: Publication
full work available at URL: s0012-365x(01)00412-5
OpenAlex ID: W2030279030
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description: scientific article; zbMATH DE number 1838905