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Keywords:

  • hypergraph;
  • finite projective plane;
  • upper chromatic number;
  • double blocking set;
  • MSC2000 Subject Classification: 05C15;
  • 05B25

Abstract

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