AbstractA set of points, S⊆PG(r, q), is said to be ϱ -saturating if, for any point x∈PG(r, q), there exist ϱ+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,ϱ ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3 q+ 1 forq= 2, q≥ 4. We further give an upper bound onk (ϱ+ 1, pm, ϱ)