On sets of sets mutually intersecting in exactly one element

Let be a finite set, and letS be a set of subsets of mutually intersecting in exactly one element, such that all elements ofS have cardinality at mostr, and such that each element of is contained in at least two elements ofS.We give an upper bound for the cardinality of and a characterization of the pairs (,S) attaining the upper bound.The problem is equivalent to determining the maximum number of lines of a linear space having at mostr lines through a point.