Saturating sets in projective planes and hypergraph covers

TL;DR

This paper improves bounds on the smallest saturating sets in finite projective planes using probabilistic and algorithmic methods, revealing connections with hypergraph transversal numbers.

Contribution

It introduces new upper bounds for saturating sets in projective planes and links these bounds to hypergraph transversal concepts using probabilistic and algorithmic approaches.

Findings

Improved upper bounds on saturating set sizes

Connection established between saturating sets and hypergraph transversal numbers

Algorithmic methods support probabilistic bounds

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.