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The 2‐Blocking Number and the Upper Chromatic Number of PG (2, q )
Author(s) -
Bacsó Gábor,
Héger Tamás,
Szőnyi Tamás
Publication year - 2013
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21347
Subject(s) - mathematics , combinatorics , blocking set , projective plane , partition (number theory) , finite field , finite set , plane (geometry) , chromatic scale , discrete mathematics , line (geometry) , projective space , geometry , projective test , pure mathematics , collineation , mathematical analysis , correlation
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.