Saturating sets in projective planes and hypergraph covers

Let Πq be an arbitrary finite projective plane of order q. A subset S of its points is called saturating if any point outside S is collinear with a pair of points from S. Applying probabilistic tools we improve the upper bound on the smallest possible size of the saturating set to ⌈3qlnq⌉+⌈(q+1)∕2⌉. The same result is presented using an algorithmic approach as well, which points out the connection with the transversal number of uniform multiple intersecting hypergraphs.