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.

翻译：线性饱和数 $sat^{lin}_k(n,\mathcal{F})$（线性极值数 $ex^{lin}_k(n,\mathcal{F})$）是指：在包含$n$个顶点的线性$k$-一致超图中，不包含$\mathcal{F}$中任何成员作为子图，但添加任意一条新超边（且结果超图仍为线性$k$-一致超图）后必产生$\mathcal{F}$中某超图的副本时，超边数的最小（最大）值。确定$ex_3^{lin}(n$, Berge-$C_3$)等价于著名的(6,3)-问题，该问题已于1976年解决。此后，Berge环的线性极值数问题得到广泛研究。作为饱和问题中该问题的对偶，本文考虑Berge环线性饱和数的确定问题。我们证明：对任意整数$k\ge3$、$t\ge 3$，有$sat^{lin}_k$($n$, Berge-$C_t)\ge \big\lfloor\frac{n-1}{k-1}\big\rfloor$，且当$t=3$时等号成立。此外，我们给出$sat^{lin}_3(n,$ Berge-$C_4)$的上界，并对任意非连通的Berge-$C_4$-饱和线性3-一致超图，给出其超边数的下界。