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Connected covering numbers

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 Turán 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.