A combinatorial game I am studying has given rise to the following question. Consider the group Z/nZ. What is the largest m such that there exist k sets m residues such that the intersection of a translation of each of these sets has at most 1 element? That is, if the sets are A_1, ..., A_m, we require that for all (c_1, ..., c_m) the intersection of (A_i + c_i) for 1 <= i <= m has at most one element, where A_i + c_i is set obtained by adding c_i to every element in A_i. Alternatively, if it makes the answer simpler, we can ask what the smallest n is given m and k.

For k=2, the answer is simple. It is possible when n>=m^2-m+1 by making one set {0,1,...,m-1} and the other {0,m,...,m(m-1)}. But it's not clear to me how to extend the construction to k>=3.

If there is a simple generalization to the case where the sets of residues can be different sizes, I'd be interested in that as well. That is, we are given k and (m_1, ..., m_k), where the ith set must have size m_i, and we are to find the smallest n for which the sets can have a single mutual intersection.

Thanks in advance for any help.