An extremal problem of d permutations containing every permutation of every t elements

Let dt(n) be the smallest number d having d permutations {σ1, σ2, …, σd} of [n] such that for every permutation τ of every t elements of [n], there exists a σi(1 ⩽ i ⩽ d) containing τ. For fixed t ⩾ 4 and large n, we show dt ⩾ (t − 2)! logt n.