Given a finite set, $A \subseteq \mathbb{R}^2$, and a subset, $B \subseteq A$, the \emph{MST-ratio} is the combined length of the minimum spanning trees of $B$ and $A \setminus B$ divided by the length of the minimum spanning tree of $A$. The question of the supremum, over all sets $A$, of the maximum, over all subsets $B$, is related to the Steiner ratio, and we prove this sup-max is between $2.154$ and $2.427$. Restricting ourselves to $2$-dimensional lattices, we prove that the sup-max is $2.0$, while the inf-max is $1.25$. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than $1.25$.

翻译：给定有限集合$A \subseteq \mathbb{R}^2$及其子集$B \subseteq A$，定义最小生成树比率（MST-ratio）为$B$与$A \setminus B$的最小生成树总长度除以$A$的最小生成树长度。关于所有集合$A$上所有子集$B$的最大值的上确界问题，与斯坦纳比率相关，我们证明该上确界值介于$2.154$与$2.427$之间。将问题限制于二维格点时，我们证明该上确界值为$2.0$，而下确界极值为$1.25$。这些结果中最为困难的是下确界极值的上界证明，我们通过证明六边形格点的MST-ratio不超过$1.25$来达成此目标。