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, 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.
For a finite projective plane , letχ( ) denote the maximum number of classes in a partition of the point set, such that each line has at least two points in the same partition class. We prove that the best possible general estimate in terms of the order of projective planes is q 2 − q − ( √ q), which is tight apart from a multiplicative constant in the third term √ q:(1) As q → ∞,χ( ) ≤ q 2 − q − √ q/2 + o( √ q) holds for every projective plane of order q.
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