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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 1, pp. 43-53 (2002)
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On the Smallest Minimal Blocking Sets in Projective Space Generating the Whole Space

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Martin Bokler, Klaus Metsch

Mathematisches Institut, Universität Giessen, Arndtstr. 2, D-35392 Giessen, e-mail: Klaus.Metsch@math.uni-giessen.de

**Abstract:** It was conjectured that the smallest minimal point sets of PG$(2s,q)$, $q$ a square, that meet every $s$-subspace and that generate the whole space are Baer subgeometries PG$(2s,\sqrt q)$. This was shown in 1971 by Bruen for $s=1$, and by Metsch and Storme [MS] for $s=2$. Our main interest in this paper is to prepare a possible proof of this conjecture by proving a strong theorem on line-blocking sets in projective spaces (see Theorem 1.1). We apply this theorem to prove the conjecture in the case $s=3$. The general case will be handled in a forthcoming paper by the first author.

[MS] K. Metsch; L. Storme: $2$-blocking sets in PG$(n,q)$, $q$ square. Beitr. Algebra Geom., submitted.

**Keywords:** smallest minimal point set; Baer subgeometry; line-blocking sets in projective space

**Classification (MSC2000):** 51E20; 05B05

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