Way to solve 5-dimentional Linear assignment problem I faced a problem that looks very hard, the problem is this:

We have 5 people putting dots on a whiteboard, each person uses colored dots with a different color (there are 5 colors, 1 for each person). My job is to  find a cluster of points which contains 3 or more distinct colors. Also, we have some fixed radius $R$, and the distance for each point in a given cluster shouldn't be more than R. Also, there might be points that don't belong to any cluster.

What kind of approach can you suggest? At first, I tried the Multidimensional linear assignment problem, but it seems that this problem is NP-hard, so I want to solve it in a reasonable amount of time (I write code). What kind of solutions can you suggest? Maybe using the fact that this isn't a general problem, I mean we use 2D space where people take action, also there are only 5 people.
If you had as much people as number of people in your clusters (each people must have a dot in each cluster) then maybe you could hope for some divide-and-conquer strategy with polynomial runtime, because of greedy-like regularities. For example, in 1D and with 3 people only, after sorting the points along the axis a greedy sweeping algorithm yields an optimal solution ; with some tweaks maybe this kind of ideas could work in higher dimensions. But with some people who can not be part of a specific cluster, this doesn't seem to work anymore.
If I were you, I'd look for approximation algorithms (good ones are known for the 3d matching). Maybe some of them can yield good approximation factors with your problem's structure.