Linearity of saturation for Berge hypergraphs

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Abstract

For a graph F, we say a hypergraph H is a Berge-F if it can be obtained from F by replacing each edge of F with a hyperedge containing it. We say a hypergraph is Berge-F-saturated if it does not contain a Berge-F, but adding any hyperedge creates a copy of a Berge- F. The k-uniform saturation number of Berge-F, satk(n,Berge-F) is the fewest number of hyperedges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show that satk(n,Berge-F)=O(n) for all graphs F and uniformities 3k5, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraphs.

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1

Research supported by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office – NKFIH, Hungary, grant K 116769.

2

Research supported by National Research, Development and Innovation Office – NKFIH, Hungary, grant K 116769.

3

Supported in part by NSF, USA grant DMS-1606350.