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arXiv · 2503.14918

Intersecting hypergraphs with large cover number

Matija Bucić·Vanshika Jain·Varun Sivashankar
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Abstract

In their famous 1974 paper introducing the local lemma, Erdős and Lovász posed a question-later referred 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.

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Matija Bucić, Vanshika Jain, Varun Sivashankar. 2025-04-14. Intersecting hypergraphs with large cover number. https://arxiv.org/abs/2503.14918

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