On multi-coloured lines in a linear space

We prove that if X₁, X₂,…, X_k are pairwise disjoint sets of points in a linear space, each of cardinality n, whose union ∪_{j = 1}^kX_j, is not collinear, then there are at least (k ? 1) n lines in the space, which intersect at least two of th e X_j's. Equality occurs if and only if k = n + 1 and X₁,…, X_{n + 1} are obtained by taking n + 1 concurrent lines in a projective plane of order n, and omitting, from each of them, their common point. When n = 1, this reduces to a theorem of de Bruijn and Erdos (Indag. Math.10 (1984), 421–423).