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

Gábor Bacsó, Tamás Héger, Tamás Szőnyi
2013 Journal of combinatorial designs (Print)  
A 2-fold 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 τ 2 (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 τ 2 (PG(2, q)) ≤ 2(q + (q − 1)/(r − 1)). For a finite projective plane Π,
more » ... 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 thatχ(Π) ≥ v − τ 2 (Π) + 1 (⋆) for every plane Π on v points. Let q = p h , p prime. We prove that for Π = PG(2, q), equality holds in (⋆) if q and p are large enough.
doi:10.1002/jcd.21347 fatcat:fkalrojogjcapn246msvlsy7vq