An Improved Lower Bound for the Erd\H{o}s-Lov\'asz Cover Number Problem

Let $g(r)$ be the minimum number of edges in an $r$-uniform intersecting hypergraph with cover number $r$.

Erdős and Lovász proved the lower bound $g(r)\ge 8r/3-3$.

We first give a completely elementary proof that $g(r)\ge 3r-4$.

We then build on the same approach and apply Kahn's small-codegree hypergraph edge-colouring theorem to improve this to $g(r)\ge (61/20-o(1))r$.

To the best of our knowledge, this is the first improvement over the Erdős-Lovász lower bound in about fifty years.