Embedding the complement of a minimal blocking set in a projective plane

✍ Scribed by Lynn Margaret Batten

• Publisher: Elsevier Science
• Year: 1984
• Tongue: English
• Weight: 373 KB
• Volume: 52
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(84)90099-2

✦ Synopsis

A blocking set B in a projective plane z of order n is a subset of T which meets every line but contains no line completely. Hence le)B n I] srz for every line i of 9r.I A blocking set is minimal if it contains no proper blocking set. A blocking set is maximal if it is not properly contained in any bk&ing set. Let B" be the complement of B in 7~. So B is a blocking set if and only if B' is lllaxgnal.

Our main thxcem is Theorem 1 below, which gives sufkient conditions for the embpd&bility of a finite linear space in a projective plane. In particular, this result generalizes the embedding of the complement of a unital [3] and the embedding of the complement of a Baer subplane [2, p. 3171. We show, as corollaries to this result, that the structure of a minimal blocking set in a finite projective plane is uniquely determined by that of its complement, and that a blocking set in a finite projective plane cannot be both maximal and mimmal. A proof of this iast-mentioned result already appears in [6:63-_ostensibly for 1;,'G@, q), but the proof is valid in any projective plan?. A linear spclce is a pair S = (P, L) where F is a set of LnGnts and L a set of lines, each line being a set of at least two points, such that each pair of points is in (on) a unique line. S is hivial if it has at most one line. IL& v = ]P], 6 = IL. S is finite if u is finite. *This research was supported in part by a grant from the University of Winnipeg and by NSERC grant T2006. ** The author is presently on sabbatical at We&eld College, University of London, and wishes to thank the College and particukly the Department of Mathematics for its hospitality.