The linear saturation number (linear extremal number ) of is the minimum (maximum) number of hyperedges of an -vertex linear -uniform hypergraph containing no member of as a subgraph, but the addition of any new hyperedge such that the result hypergraph is still a linear -uniform hypergraph creates a copy of some hypergraph in . Determining , Berge-) 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 (, Berge- for any integers , , and the equality holds if . In addition, we provide an upper bound for and for any disconnected Berge--saturated linear 3-uniform hypergraph, we give a lower bound for the number of hyperedges of it.

中文翻译：

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Berge-C3 和 Berge-C4 的线性饱和数

的线性饱和数（线性极值数 ）是不包含 的成员的 顶点线性均匀超图的最小（最大）超边数 作为子图，但添加任何新的超边使得结果超图仍然是线性均匀超图在 中创建某些超图的副本。确定 , Berge-) 相当于著名的 (6,3)-问题，该问题于 1976 年得到解决。从那时起，确定 Berge 循环的线性极值数就得到了广泛的研究。作为该问题在饱和问题中的对应问题，考虑确定伯格循环的线性饱和数的问题。在本文中，我们证明 (, Berge- 对于任何整数 , ，并且如果 则等式成立。此外，我们提供了一个上限，并且对于任何不连通的 Berge——饱和线性 3-均匀超图，我们给出了一个下界为它的超边数。