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

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 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, letχ ¯ ( 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.