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On sets of points with few odd secants

Part of: Finite geometry and special incidence structures

Published online by Cambridge University Press:  10 October 2019

Simeon Ball  and
Bence Csajbók
Simeon Ball*
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Mòdul C3, Campus Nord, c/ Jordi Girona 1–3, 08034 Barcelona, Spain
Bence Csajbók
Affiliation:
MTA–ELTE Geometric and Algebraic Combinatorics Research Group, ELTE Eötvös Loránd University, Budapest, Hungary, Department of Geometry, 1117 Budapest, Pázmány P. stny. 1/C, Hungary
*
*Corresponding author. Email: simeon@ma4.upc.edu
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Abstract

We prove that, for q odd, a set of q + 2 points in the projective plane over the field with q elements has at least 2q − c odd secants, where c is a constant and an odd secant is a line incident with an odd number of points of the set.

MSC classification

Primary: 51E20: Combinatorial structures in finite projective spaces
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 1 , January 2020 , pp. 31 - 43
Copyright
© Cambridge University Press 2019 

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Footnotes

†

The first author acknowledges the support of project MTM2017-82166-P of the Spanish Ministerio de Economía y Competitividad.

‡

The second author is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The second author acknowledges the support of OTKA grant K 124950.

References

Balister, P., Bollobás, B., Füredi, Z. and Thompson, J. (2014) Minimal symmetric differences of lines in projective planes. J. Combin. Des. 22 435–451.CrossRefGoogle Scholar
Ball, S. (2012) On sets of vectors of a finite vector space in which every subset of basis size is a basis. J. Eur. Math. Soc. 14 733–748.CrossRefGoogle Scholar
Ball, S. and Lavrauw, M. (2018) Planar arcs. J. Combin. Theory Ser. A 160 261–287.CrossRefGoogle Scholar
Bichara, A. and Korchmáros, G. (1982) Note on (q + 2) -sets in a Galois plane of order q. Ann. Discrete Math. 14 117–122.Google Scholar
Blokhuis, A. (1991) Characterization of seminuclear sets in a finite projective plane. J. Geom. 40 15–19.CrossRefGoogle Scholar
Blokhuis, A. and Bruen, A. A. (1989) The minimal number of lines intersected by a set of q + 2 points, blocking sets, and intersecting circles. J. Combin. Theory Ser. A 50 308–315.CrossRefGoogle Scholar
Blokhuis, A. and Mazzocca, F. (2008) The finite field Kakeya problem. In Building Bridges, Vol. 19 of Bolyai Soc. Math. Stud., Springer, pp. 205–218.Google Scholar
Blokhuis, A., Seress, A. and Wilbrink, H. A. (1991) On sets of points without tangents. Mitt. Math. Sem. Univ. Giessen 201 39–44.Google Scholar
Csajbók, B. (2018) On bisecants of Rédei type blocking sets and applications. Combinatorica 38 143–166.CrossRefGoogle Scholar
Segre, B. (1955) Ovals in a finite projective plane. Canad. J. Math. 7 414–416.CrossRefGoogle Scholar
Segre, B. (1967) Introduction to Galois geometries. Atti Accad. Naz. Lincei Mem. 8 133–236.Google Scholar
Szönyi, T. and Weiner, Z. (2014) On the stability of the sets of even type. Adv. Math. 267 381–394.Google Scholar
Vandendriessche, P. (2015) On small line sets with few odd-points. Des. Codes Cryptogr. 75 453–463.CrossRefGoogle Scholar