co.combinatorics - partition of a set

First, we see this example.
Suppose we have a set of 6 elements, we can get 3 subsets of it, each of which has 2 elements, but no two sets overlap.
But if our set has 5 elements, we want to get 3 subsets of it each of which has 2 elements
. Then two of them need to overlap on 1 element.
Now generally suppose we have a set of #a elements and we want 3 subsets of it each of which has #a' elements.
We also want each two of them have the same but least # in common.
Could you get the relationship between a and a'? How many does any two of them in common?
How many does three of them has in common?
For example, if a=3a', the we can have no pair of the three subsets overlap.