Covering by intersecting families

A set family A⊂[n](k) is called noncentred intersecting if it is intersecting but ⋂A∈AA=∅; let Lk(n) be the least number of noncentred intersecting families which cover [n](k). We prove that for k⩾3 fixed,Lk(n)=n22k(k−1)1+O1n.