Estimates of antipodal sets in oriented real Grassmann manifolds

We estimate the cardinalities of antipodal sets in oriented real Grassmann manifolds of low ranks. The author reduced the classification of antipodal sets in oriented real Grassmann manifolds to a certain combinatorial problem in a previous paper. So we can reduce estimates of the antipodal sets to those of certain combinatorial objects. The sequences of antipodal sets we obtained in previous papers show that the estimates we obtained in this paper are the best.