A Design Type Result Via Hamiltonia Cycles

Let X be a finite set of n elements and be integers. Unordered pairs of disjoint q-element subsets of X will be considered. We say that two such pairs are far if implies and implies . We will prove that if n is large enough then the family of all q-element subsets of X with one possible exception (if the number of q-element subsets is odd) can be partitioned into disjoint pairs of q-element subsets which are pairwise far. The proof uses a generalization of Dirac's theorem on Hamiltonian cycles.