Take a set of $2n$ points in the plane and assume that no open set of diameter $1$ contains more than $n$ of these points. 


Question: can we pair up the points so that the distance between the points in a pair is at least $1$? 



On distances between points on the plane
In Gjergi's $n=2$ counterexample, the center is an isolated point of the graph you'd like to match. One way to save your question might simply rule that out.


But consider your question *without* the hypothesis: When can we pair up $2n$ points so that the distance between the points in a pair is at least 1? General matching theory says yes, one can, whenever there is no Tutte obstruction. That means whenever you can't remove $k$ points and leave a graph with more than $k$ odd components. Now for two points to lie in distinct components is a strong condition: no path with big steps can connect them. In particular the distance between the points cannot be more than 1, and they can't both lie more than 1 unit away from any common third point.


Now one can get many components of size 1 by having many points clustered close together, which is roughly what your hypothesis meant to rule out. But even arranging points along a line, one can get many components and not all of them consisting of isolated points.


Hope that helps...sorry, don't have time to work this out further now.


