Covers in Uniform Intersecting Families and a Counterexample to a Conjecture of Lovلsz

We discuss the maximum size of uniform intersecting families with covering number at leastτ. Among others, we construct a largek-uniform intersecting family with covering numberk, which provides a counterexample to a conjecture of Lovász. The construction for oddkcan be visualized on an annulus, while for evenkon a Möbius band.