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 $k$. ErdH{o}s and Lovasz proved that $ lfloor k! (e-1) rfloor leq r(k) leq k^k.$ Frankl, Ota, and Tokushige improved the lower bound to $r(k) geq left( k/2 right)^{k-1}$, and Tuza improved the upper bound to $r(k) leq (1-e^{-1}+o(1))k^k$. We establish that $ r(k) leq (1 + o(1)) k^{k-1}$.