Partition n objects into k groups with required repeats and pairwise restrictions

Question

Is there a way to partition $ n $ objects into $ k $ groups, with repeats, with the following requirements:

1. each object should appear in at least $ m $ groups
2. no two objects can be grouped together more than $ p $ times

How do we think about the grouping strategy and minimizing the number of groups? Is this a known/solved problem? In the worst case, I'm also open to a procedural solution in pseudocode (brute force is fine)