Abstract A set S of vertices in a hypergraph H is a transversal if it has a nonempty intersection with every edge of H . For k ≥ 1 , if H is a hypergraph with every edge of size at least k , then a k -transversal in H is a transversal that intersects every edge of H in at least k vertices. In particular, a 1-transversal is a transversal. The upper k -transversal number ϒ k ( H ) of H is the maximum cardinality of a minimal k -transversal in H . We obtain asymptotically best possible lower bounds on ϒ k ( H ) for uniform hypergraphs H . More precisely, we show that for r ≥ 2 and for every integer k ∈ [ r ] , if H is a connected r -uniform hypergraph with n vertices, then ϒ k ( H ) > 2 3 n r − k + 1 . For r > k ≥ 1 and e > 0 , we show that there exist infinitely many r -uniform hypergraphs, H r , k ∗ , of order n and a function f ( r , k ) of r and k satisfying ϒ k ( H r , k ∗ ) ( 1 + e ) ⋅ f ( r , k ) ⋅ n r − k + 1 .