  Let $\Pi_q$ 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
$\lceil\sqrt{3q\ln{q}}\rceil+ \lceil(\sqrt{q}+1)/2\rceil$. 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.
