Linearity of saturation for Berge hypergraphs

On saturation of Berge hypergraphs

“…Here and throughout the paper the parameter r in the index denotes that we consider r-uniform hypergraphs, and we will denote sat r (n, Berge-F ) by sat r (n, F ) for brevity. The conjecture was proved for 3 ≤ r ≤ 5 and any F in [4]. In this paper we gather some further results that support the conjecture.…”

“…English, Gerbner, Methuku and Tait [4] extended this conjecture to hypergraph-based Berge hypergraphs. Analogously to the graph-based case, we say that a hypergraph H is a Berge copy of a hypergraph F (in short: H is a Berge-F ) if V (F ) ⊂ V (H) and there is a bijection f : E(F ) → E(H) such that for any e ∈ E(F ) we have e ⊂ f (e).…”

Saturation Number of Berge Stars in Random Hypergraphs

“…We say an r-uniform hypergraph H is a Berge-G if there exists a bijection φ : E(G) → E(H) such that e ⊆ φ(e) for each edge e ∈ E(G). Recently, extremal problems for Berge hypergraphs have attracted the attention of a lot of researchers, see, e.g., [3,2,4,5,8,13]. In 2018, Austhof and English [2] studied the saturation number of Berge stars.…”

“…for large n, which generalizes equation (1) to uniform hypergraphs. In 2019, English et al [5] proved that sat(n, Berge-F ) = O(n) for any graph F and uniformities 3 r 5.…”

A Note on Saturation for Berge-G Hypergraphs

“…This result is asymptotically tight for some G. The determination of a smallest size for {G}-saturated hypergraphs remains open in general. In the same setting of k-uniform hypergraphs, English et al [2] proved that there are B k (G) saturated hypergraphs on n vertices and O(n) hyperedges, where B k (G) is the set of all k-uniform Berge-G hypergraphs, 3 ≤ k ≤ 5. See also English et al [3], for Berge saturation results on some special graphs.…”