For k = (k 1 , · · · , kn) ∈ N n , 1 ≤ k 1 ≤ · · · ≤ kn, let L r k be the family of labeled r-sets on k given by L r k := {{(a 1 , la 1 ), · · · , (ar , la r )} : {a 1 , · · · , ar} ⊆ [n], la i ∈ [ka i ], i = 1, · · · , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.