The Erdős-Sós conjecture states that the maximum number of edges in an -vertex graph without a given -vertex tree is at most . Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a -vertex tree , we construct -vertex connected graphs that are -free with at least edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of -vertex brooms such that the maximum size of an -vertex connected -free graph is at most .

中文翻译：

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连通性如何影响树的极值数量

Erdős-Sós 猜想指出，没有给定顶点树的顶点图中的最大边数最多为 。尽管人们对此很感兴趣，但这个猜想仍然悬而未决。最近，Caro、Patkós 和 Tuza 考虑了连接的主机图的这个问题。解决他们提出的问题，对于 - 顶点树 ，我们构造至少具有边的 - 自由的 - 顶点连通图，表明附加连通性条件最多可以将最大尺寸减少 2 倍。此外，我们表明这是最优的：存在一系列 - 顶点扫帚，使得 - 顶点无连通图的最大尺寸至多为 。