Linear spaces with small generated subspaces

The dimension of a linear space is the maximum positive integer d such that any d of its points generate a proper subspace. Given n, d, s, we consider linear spaces on n points such that any d points generate subspaces of size at most s. Certain design-theoretic constructions and applications are investigated. In particular, one consequence is the existence of proper n-edge-colourings of both K n + 1 (for n odd) and K n , n with a constant bound on the length of two-colored cycles.