Let $P$ be a generic set of $n$ points in the plane, and let $P=R\cup B$ be a coloring of $P$ in two colors. We are interested in the number of crossings between the minimum spanning trees (MSTs) of $R$ and $B$, denoted by $\crossAB(R,B)$. We define the \emph{bicolored MST crossing number} of $P$, denoted by $\cross(P)$, as $\cross(P) = \max_{P= R\cup B}(\crossAB(R,B))$. We prove a linear upper bound for $\cross(P)$ when $P$ is generic. If $P$ is dense or in convex position, we provide linear lower bounds. Lastly, if $P$ is chosen uniformly at random from the unit square and is colored uniformly at random, we prove that the expected value of $\crossAB(R,B)$ is linear.

翻译：设 $P$ 为平面上 $n$ 个点的一般位置集合，且设 $P=R\cup B$ 为 $P$ 的一种二染色方案。我们关注 $R$ 与 $B$ 的最小生成树（MSTs）之间的相交次数，记为 $\crossAB(R,B)$。我们定义 $P$ 的\emph{双色最小生成树相交数} $\cross(P)$ 为 $\cross(P) = \max_{P= R\cup B}(\crossAB(R,B))$。我们证明了当 $P$ 处于一般位置时，$\cross(P)$ 具有线性上界。若 $P$ 是稠密的或处于凸位置，我们给出了线性下界。最后，若 $P$ 从单位正方形中均匀随机选取并均匀随机染色，我们证明了 $\crossAB(R,B)$ 的期望值是线性的。