The minimum number of vertices in uniform hypergraphs with given domination number

The domination number γ(H) of a hypergraph H = (V (H), E(H)) is the minimum size of a subset D ⊂ V (H) of the vertices such that for every v ∈ V (H) \D there exist a vertex d ∈ D and an edge H ∈ E(H) with v, d ∈ H. We address the problem of finding the minimum number n(k, γ) of vertices that a k-uniform hypergraph H can have if γ(H) ≥ γ and H does not contain isolated vertices. We prove that n(k, γ) = k + Θ(k1−1/γ) and also consider the s-wise dominating and the distance-l dominating version of the problem. In particular, we show that the minimum number ndc(k, γ, l) of vertices that a connected k-uniform hypergraph with distance-l domination number γ can have is roughly kγl 2 .