Intersecting hypergraphs with large cover number

Abstract

In their famous 1974 paper introducing the local lemma, Erd\H{o}s and Lov\'asz posed a question-later referred by Erd\H{o}s 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 Bollob\'as' ''power of many colours'' problem, Alon, Buci\'c, 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.