A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has ${\mathcal{O}}(n^{2/({\ell}+1)})$ edges. Finally, we show that $\Theta(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.

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兼容匹配

如果匹配在每个点集上的直线绘图不交叉，则它们与两个或更多个大小为$ n $且带有标签$ \ {1，\ dots，n \} $的标记点集兼容。我们研究了与平面中一般位置上的两个或多个标记点集兼容的匹配中的最大边数。我们表明，对于$ n $点的任何两个标记凸集，存在与$ \ lfloor \ sqrt {2n} \ rfloor $边的兼容匹配。更一般而言，对于任何带有$ \ ell $标签的点集，我们构造大小为$ \ Omega（n ^ {1 / \ ell}）$的兼容匹配。作为一个相应的上限，我们使用概率论证表明，对于给定的$ n $个点集，每个$ \ ell $都存在每个集合的标签，使得最大的兼容匹配项具有$ {\ mathcal {O}}（n ^ {2 /（{{ell} +1）}）$条边。最后，