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 isroughly kγl2.