Let X be an algebraic variety over the algebraically closed field K and Ξ ⊆ X(K) a set of K-rational points on X. We say that a subvariety Y ⊆ Xn of some Cartesian power of X is Ξ-special if Ξn ∩ Y (K) is Zariski dense in Y . We show under certain hypotheses on Ξ, for instance, that the class of Ξ-special varieties is closed under intersections, that when subvarieties of X vary in an algebraic ...