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The 2-Blocking Number and the Upper Chromatic Number of PG(2,q)


  • 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.).


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 math formula 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 math formula (⋆) for every plane Π on v points. Let math formula, p prime. We prove that for math formula, equality holds in (⋆) if q and p are large enough.