Blocking Sets in Small Finite Linear Spaces

Abstract

We classify all finite linear spaces on at most \(15\) points admitting a blocking set. There are no such spaces on \(11\) or fewer points, one on \(12\) points, one on \(13\) points, two on \(14\) points, and five on \(15\) points. The proof makes extensive use of the notion of the weight of a point in a \(2\)-coloured finite linear space, as well as the distinction between minimal and non-minimal \(2\)-coloured finite linear spaces. We then use this classification to draw some conclusions on two open problems on the \(2\)-colouring of configurations of points.