Inequalities for minimal covering sets in set systems of given rank (Q1329820)

The structure of \(\nu\)-critical graphs is determined in a well-known paper of Gallai. For \(\nu\)-critical hypergraphs of rank \(\geq 3\) there are some weaker results due to P. Erdős and L. Lovász. These results were generalized by Lovász. In this paper, a sharper version of Lovász's theorem is proved, showing that if \(\mathbb{F}\) is a \(\nu\)-critical hypergraph of rank \(\geq 3\), \(\nu(\mathbb{R})= \nu\), then \(|\mathbb{F}|\leq (1- e^{-1}+ \varepsilon_ r)\) \((\nu r)^ \nu\), where \(\varepsilon_ r\) is real, \(0< \varepsilon_ r< e^{-1}\), such that \(\varepsilon_ r\to 0\) as \(r\to\infty\). The proof is driven through powerful inequalities involving the systems of the minimal transversals of \(\mathbb{F}\).