### 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 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, 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.

Original language | English |
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Journal | Journal of Combinatorial Designs |

DOIs | |

Publication status | Accepted/In press - 2013 |

### Fingerprint

### Keywords

- 05B25
- Double blocking set
- Finite projective plane
- Hypergraph
- MSC2000 Subject Classification: 05C15
- Upper chromatic number

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Combinatorial Designs*. https://doi.org/10.1002/jcd.21347

**The 2-Blocking Number and the Upper Chromatic Number of PG(2,q).** / Bacsó, Gábor; Héger, Tamás; Szőnyi, T.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The 2-Blocking Number and the Upper Chromatic Number of PG(2,q)

AU - Bacsó, Gábor

AU - Héger, Tamás

AU - Szőnyi, T.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - 05B25

KW - Double blocking set

KW - Finite projective plane

KW - Hypergraph

KW - MSC2000 Subject Classification: 05C15

KW - Upper chromatic number

UR - http://www.scopus.com/inward/record.url?scp=84875925395&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875925395&partnerID=8YFLogxK

U2 - 10.1002/jcd.21347

DO - 10.1002/jcd.21347

M3 - Article

JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

SN - 1063-8539

ER -