On sets of sets mutually intersecting in exactly one element

Let Ω be a finite set, and let S be a set of subsets of Ω mutually intersecting in exactly one element, such that all elements of S have cardinality at most r, and such that each element of Ω is contained in at least two elements of S. We give an upper bound for the cardinality of Ω and a characterization of the pairs (Ψ, S) attaining the upper bound. The problem is equivalent to determining the maximum number of lines of a linear space having at most r lines through a point. © Birkhäuser Verlag, Basel, 2000.