This problem is relevant to determining the number of discriminable combinations of components in a sensory perception task.
Suppose that there are N components to choose from, and we are only concerned with combinations of k of those components, so that there are C(N,k) combinations. I would like to find the size s of the largest set of those combinations such that each combination in the set shares no more than d components with each other combination in the set. For example, given N=6, k=3, the C(6,3)=20 combinations are:
{1,2,3},{1,2,4},{1,2,5},{1,2,6}, {1,3,4},{1,3,5},{1,3,6}, {1,4,5},{1,4,6}, {1,5,6}, {2,3,4},{2,3,5},{2,3,6}, {2,4,5},{2,4,6}, {2,5,6}, {3,4,5},{3,4,6}, {3,5,6}, {4,5,6}
And the largest set such that all members have no more than d=1 shared component is:
{{1,2,3},{1,4,5},{2,5,6},{3,4,6}}, which has size s=4. If any other combinations were added they would have more than 1 shared component with one of those 4. There are also other sets of size 4 with this property among those combinations, but none of size 5.
I've also noticed that, in that example, those 4 combinations can be represented as 4 vertices in a hexeract (6-cube), forming a tetrahedron. I don't know if the best approach to this problem is geometrical or set-theoretical, but either way it is beyond me.
To summarize, I am looking for a function for s of the form s = f(N,k,d).