Dimensions of strong n-point sets

n-point sets (plane sets which hit each line in n points) and strong n-point sets (in addition hit each circle in n-points) exist (for n ⩾ 2 , n ⩾ 3 respectively) by transfinite induction, but their properties otherwise are difficult to establish. Recently for n-point sets the question of their possible dimensions has been settled: 2- and 3-point sets are always zero-dimensional, while for n ⩾ 4 , one-dimensional n-point sets exist. We settle the same question for strong n-point sets: strong 4- and 5-point sets are always zero-dimensional, while for n ⩾ 6 , both zero-dimensional and one-dimensional strong n-point sets exist.