My friend (who is a medical student!) posed me the following question:

There are 70 people, and you want to split them up into 10 groups of 7 people each. Two such partitions are "compatible" if no two individuals are together in the same group in both. What is the maximal number of compatible partitions you can form?

There is an obvious generalization here. When the number of groups is prime, there seems to be a pretty efficient way of generating lots of compatible partitions by rotating people in different increments. However, even though this particular case is in principle a ``finite computation,'' I don't see a feasible way to compute the answer.