This paper studies the cardinality of a smallest set $ {\user1{T}} $ {\user1{T}} of t-subspaces of the finite projective spaces PG(n, q) such that every s-subspace is incident with at least one element of $ {\user1{T}} $ {\user1{T}} , where 0 t < s n. This is a very difficult problem and the solution is known only for very few families of triples (s, t, n). When the answer is known, the corresponding blocking configurations usually are partitions of a subspace of PG(n, q) by subspaces of dimension t. One of the exceptions is the solution in the case t = 1 and n = 2s. In this paper, we solve the case when t = 1 and 2s < n 3s-3 and q is sufficiently large.