Dual blocking sets in projective and affine planes

✍ Scribed by Peter J. Cameron; Francesco Mazzocca; Roy Meshulam

• Publisher: Springer
• Year: 1988
• Tongue: English
• Weight: 191 KB
• Volume: 27
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00151351

✦ Synopsis

A dual blocking set is a set of points which meets every blocking set but contains no line. We establish a lower bound for the cardinality of such a set, and characterize sets meeting the bound, in projective and affine planes.

A blocking set for a family ~-of sets is a set which meets every member of~ but contains none. Blocking sets have been studied intensively, especially in the case where Y is the set of lines of a projective or affine plane (see, for example, [1]). Two of the motivating questions are: What is the minimum size of a blocking set? and What is the structure of blocking sets of minimal size?

A dual blocking set for ~ is a set which meets every blocking set for ~ but contains no member of ~. In the course of showing that a projective or affine plane is, in general, determined by its family of blocking sets, the first two authors showed in [2] that, for such planes, the smallest sets meeting every blocking set are the lines: in other words, a dual blocking set has larger cardinality than a line. The question 'How much larger?' was left open. We propose to answer that question here.

We require one further definition in order to state our results. A line oval in a projective plane of order n is a set of n + 2 lines, no three concurrent. A line oval in an affine plane of order n is a line oval in the corresponding projective plane, one of whose lines is the line at infinity: in other words, a set of n + 1 lines, no three concurrent and no two parallel. It is well known that line ovals exist only in planes of even order, and that any Desarguesian plane of even order contains them.

Projective planes of order 2, and affine planes of order 2 or 3, contain no blocking sets; so we exchrde these.