Bacsó, Gábor and Héger, Tamás and Szőnyi, Tamás (2013) The 2-Blocking Number and the Upper Chromatic Number of PG(2,q). JOURNAL OF COMBINATORIAL DESIGNS, 21 (12). pp. 585-602. ISSN 1063-8539 MTMT:2274370; doi:10.1002/jcd.21347

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Abstract

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 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 τ2PG(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 ({star operator}) for every plane Π on v points. Let q=ph, p prime. We prove that for Π= PG (2,q), equality holds in ({star operator}) if q and p are large enough. © 2013 Wiley Periodicals, Inc.

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