Saturation numbers for Berge cliques

Abstract

Let F be a graph and H be a hypergraph, both embedded on the same vertex set. We say H is a Berge-F if there exists a bijection ϕ:E(F)→E(H) such that e⊆ϕ(e) for all e∈E(F). We say H is Berge-F-saturated if H does not contain any Berge-F, but adding any missing edge to H creates a copy of a Berge-F. The saturation number satk(n,Berge-F) is the least number of edges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show satk(n,Berge-Kℓ)∼ℓ−2k−1n,for all k,ℓ≥3. Furthermore, we provide some sufficient conditions to imply that satk(n,Berge-F)=O(n) for general graphs F.