Connected covering numbers

TL;DR

This paper studies connected coverings, providing formulas, bounds, and connections to Turán systems, and introduces an improved general upper bound impacting oriented matroid theory.

Contribution

It presents new formulas and bounds for connected covering numbers, and introduces an improved upper bound that enhances previous results and applications.

Findings

Derived various formulas and bounds for connected covering numbers.

Established connections between connected coverings and Turán systems.

Provided an improved upper bound impacting oriented matroid research.

Abstract

A connected covering is a design system in which the corresponding {\em block graph} is connected. The minimum size of such coverings are called {\em connected coverings numbers}. In this paper, we present various formulas and bounds for several parameter settings for these numbers. We also investigate results in connection with {\em Tur\'an systems}. Finally, a new general upper bound, improving an earlier result, is given. The latter is used to improve upper bounds on a question concerning oriented matroid due to Las Vergnas.