Saturating sets in projective planes and hypergraph covers

Abstract

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.

arXiv comment10 pages, detailed calculations are included compared to V1