Linear spaces in which every long line intersects \(v\) + 2 other lines (Q1193254)

A finite linear space, \(\text{\textbf{L}}=({\mathcal P},{\mathcal L})\), consists of a finite set \(\mathcal P\) of points and a set \(\mathcal L\) of subsets of \(\mathcal P\) called lines such that 1) any two distinct points are contained in a unique line, 2) any line contains at least two points, and 3) there are three non-collinear points. Let \(|{\mathcal P}|=v\), \(|{\mathcal L}|=b\) and \(n\) be the maximum cardinality of any line in the finite linear space. Any line of cardinality \(n\) is called a long line and all others are considered to be short. \textit{L. E. Varga} [J. Comb. Theory, Ser. A 40, 435-438 (1985; Zbl 0575.05007)] has shown that in any linear space \textbf{L}, every long line meets at least \(v\) other lines unless \textbf{L} is a projective plane or a near-pencil. The linear spaces in which the long lines intersect either \(v\) or \(v=1\) other lines have been studied by two of the authors and N. Melone at the Universitá di Napoli. The current work is an investigation of linear spaces in which every long line meets \(v+2\) other lines and \(n\geq 7\). This body of work (including the paper under review) gives credence to the conjecture: Given an integer \(x\), then, apart from a finite number of exceptions, every linear space in which every long line intersects at most \(v+x\) other lines can be obtained from an affine plane by removing at most \(x\) points or from a projective plane removing at most \(x+1\) points. In this paper it is shown that, under the given assumption, if \(r\) denotes the minimal number of lines passing through any point of \textbf{L}, then \(r=n\) or \(r=n+1\). These two cases give rise to the punctured projective and affine planes respectively, as well as a few sporadic examples.