Saturation for the 3-uniform loose 3-cycle

Abstract

Let F and H be k-uniform hypergraphs. We say H is F-saturated if H does not contain a subgraph isomorphic to F, but H+e does for any hyperedge e∉E(H). The saturation number of F, denoted satk(n,F), is the minimum number of edges in an F-saturated k-uniform hypergraph H on n vertices. Let C3(3) denote the 3-uniform loose cycle on 3 edges. In this work, we prove that(43+o(1))n≤sat3(n,C3(3))≤32n+O(1). This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.