Title:Linear saturation numbers of Berge-$C_3$ and Berge-$C_4$

Abstract:The linear saturation number $sat^{lin}_k(n,\mathcal{F})$ (linear extremal number $ex^{lin}_k(n,\mathcal{F})$) of $\mathcal{F}$ is the minimum (maximum) number of hyperedges of an $n$-vertex linear $k$-uniform hypergraph containing no member of $\mathcal{F}$ as a subgraph, but the addition of any new hyperedge such that the result hypergraph is still a linear $k$-uniform hypergraph creates a copy of some hypergraph in $\mathcal{F}$. Determining $ex_3^{lin}(n$, Berge-$C_3$) is equivalent to the famous (6,3)-problem, which has been settled in 1976. Since then, determining the linear extremal numbers of Berge cycles was extensively studied. As the counterpart of this problem in saturation problems, the problem of determining the linear saturation numbers of Berge cycles is considered. In this paper, we prove that $sat^{lin}_k$($n$, Berge-$C_t)\ge \big\lfloor\frac{n-1}{k-1}\big\rfloor$ for any integers $k\ge3$, $t\ge 3$, and the equality holds if $t=3$. In addition, we provide an upper bound for $sat^{lin}_3(n,$ Berge-$C_4)$ and for any disconnected Berge-$C_4$-saturated linear 3-uniform hypergraph, we give a lower bound for the number of hyperedges of it.