Contract grant sponsor: OTKA; contract grant number: K 81310 (T. H. and T. Sz.); contract grant sponsor: ERC; contract grant number: 227701 DISCRETECONT (T. H.); contract grant sponsor: Slovenian--Hungarian Intergovernmental Scientific and Technological Project; contract grant number TÉT-10-1-2011-0606 (T. Sz.).
The 2-Blocking Number and the Upper Chromatic Number of PG(2,q)
Article first published online: 10 APR 2013
© 2013 Wiley Periodicals, Inc.
Journal of Combinatorial Designs
Volume 21, Issue 12, pages 585–602, December 2013
How to Cite
Bacsó, G., Héger, T. and Szőnyi, T. (2013), The 2-Blocking Number and the Upper Chromatic Number of PG(2,q). J. Combin. Designs, 21: 585–602. doi: 10.1002/jcd.21347
- Issue published online: 2 OCT 2013
- Article first published online: 10 APR 2013
- Manuscript Revised: 28 FEB 2013
- Manuscript Received: 18 JUN 2012
- OTKA. Grant Number: K 81310
- ERC. Grant Number: 227701
- Slovenian--Hungarian Intergovernmental Scientific and Technological. Grant Number: TÉT-10-1-2011-0606
- finite projective plane;
- upper chromatic number;
- double blocking set;
- MSC2000 Subject Classification: 05C15;
A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ2(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ2PG(2,q))≤ 2(q+(q-1)/(r-1)). For a finite projective plane Π, let denote the maximum number of classes in a partition of the point-set, such that each line has at least two points in some partition class. It can easily be seen that (⋆) for every plane Π on v points. Let , p prime. We prove that for , equality holds in (⋆) if q and p are large enough.