For all possible non-decreasing finite sequences, $S_l$, of positive integers with values in $Z_n = \{0,...,n-1\}$ and size $|S_l|=n, l = 1,2,...,n^n$, what is the minimal size of a maximal set of disjoint triplets $[x_k, x_j, x_i], x_i, x_j, x_k \in S_l$ that satisfy the equidistant property: $x_i - x_j = x_j - x_k, 1 \le k < j < i \le n$.
For example, sequence $S = \{0,1,3,3,4,5,5,7,8\}$, with $9$ values in $Z_9$ has a maximal set of 3 disjoint triplets $[x_k, x_j, x_i]$ with equidistant property: $[(0,4,8),(1,3,5),(3,5,7)]$
I'm looking for a theory on the minimum bound of such maximal sets for all possible sets $|S_l|=n, l = 1,2,...,n^n$ made by taking $n$ values from $Z_n$ (where, obviously, some values may repeat, etc).
Is there any underlying theory for such a problem? Where would I start researching this? It looks like a problem on the border of abstract algebra, combinatorics, and number theory. Thanks in advance.
(edited for clarity and with some loss of generality)