On finite matroids with one more hyperplane than points (Q1911998)

Let \(n\) be a natural number \(\geq 2\). An \(n\)-dimensional Bridges space is a finite \(n\)-dimensional linear space in which there are \(b_{n+1} = \nu + 1\) hyperplanes, where \(\nu\) is the number of points. The main result of the paper is the following theorem: An \(n\)-dimensional Bridges space \(L\) with \(n \geq 3\) has one of the following structures: (i) an \(n\)-dimensional Galois projective space with one point deleted; (ii) the direct sum of a generalized projective space and a \(t\)-dimensional Bridges space, for a suitable \(t \leq n - 1\). This result extends the classification of 2-dimensional and 3-dimensional Bridges spaces to a general \(n\).