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 ofΠis 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 τ2(PG(2, q)) ≤ 2(q + (q - 1)/(r - 1)). For a finite projective plane Π, let χ (Π) 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 χ(Π) ≥ v - τ2(Π) + 1 (*) for every plane Π on v points. Let q = ph, p prime. We prove that for Π = PG(2, q), equality holds in (*) if q and p are large enough.
- Double blocking set
- Finite projective plane
- Upper chromatic number
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics