Intersecting hypergraphs with large cover number
Matija Bucić, Vanshika Jain, Varun Sivashankar
TL;DR
This work addresses the vertex-constrained version of Erdős–Lovász’s question on the minimum edge count in an $r$-uniform intersecting hypergraph with maximal cover number by introducing $f(n,r)$ for $n$-vertex hypergraphs. The authors develop a wreath-product construction $H_1 \rtimes H_2$ that preserves intersecting and criticality while interpolating between extremal configurations, and complement it with a uniformity-augmentation lemma to adjust the hypergraph’s uniformity. They prove that for $3\le 2r-1 \le n \le O(r^2)$, there exists an $n$-vertex $r$-uniform critical hypergraph with at most $(n/r)^{O(r^2/n)}$ edges, establishing $f(n,r) = 2^{\tilde{\Theta}(r^2/n)}$ up to a logarithmic factor in the exponent. This yields a tight asymptotic description (up to log factors) across the relevant regime and demonstrates how a product construction can smoothly interpolate between known extremal instances, with potential implications for related Turán-type and covering-design problems.
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.