The Euclidean MST-ratio for Bi-colored Lattices

Given a finite set, A ⊆ℝ^2, and a subset, B ⊆ A, the MST-ratio is the combined length of the minimum spanning trees of B and A ∖ 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.