Intersecting hypergraphs with large cover number
Published Web Location
https://doi.org/10.5070/C6.69920
Abstract
In their famous 1974 paper introducing the local lemma, Erdős and Lovász posed a question—later referred to by Erdős as one of his three favorite open problems: What is the minimum number of edges in an \( r \)-uniform, intersecting hypergraph with cover number \( r \)? This question was solved up to a constant factor in Kahn's remarkable 1994 paper. More recently, motivated by applications to Bollobás' "power of many colours" problem, Alon, Bucić, Christoph, and Krivelevich introduced a natural generalization by imposing a space constraint that limits the hypergraph to use only \(n\) vertices. In this note, we settle this question asymptotically, up to a logarithmic factor in \(n/r\) in the exponent, for the entire range.
Mathematics Subject Classifications: 05D05
Keywords: Hypergraphs, cover number, intersecting