On Compatible Matchings

. A matching is compatible to two or more labeled point sets of size n with labels { 1 , . . . , 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 sets of n points in convex position there exists a compatible matching with ⌊√ 2 n + 1 − 1 ⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω( n 1 /ℓ ). As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O ( n 2 / ( ℓ +1) ) edges. Finally, we show that Θ(log n ) copies of any set of n points are necessary and sufficient for the existence of labelings of these point sets such that any compatible matching consists only of a single edge.