An upper bound for the size of a k-uniform intersecting family with covering number k

Let r(k) denote the maximum number of edges in a k-uniform intersecting family with covering number k. Erdős and Lovász proved that ⌊k!(e−1)⌋≤r(k)≤kk. Frankl, Ota, and Tokushige improved the lower bound to r(k)≥(k/2)k−1, and Tuza improved the upper bound to r(k)≤(1−e−1+o(1))kk. We establish that r(k)≤(1+o(1))kk−1.