An Improved Lower Bound for the Erdős-Lovász Cover Number Problem

Abstract

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 $g(r)\ge ((41-\sqrt{19})/12-o(1))r\approx 3.053r$. In particular, this shows that $g(r)>3r$ for all sufficiently large $r$, addressing a question of Erdős.

arXiv commentSmall improvement in the constant