Nonhomogeneous analytic families of trees

Abstract

We consider a dichotomy for analytic families of subtrees of a treeT\mathbb {T}stating that either there is a colouring of the nodes ofT\mathbb {T}for which all but finitely many levels of every tree in the family are nonhomogeneous or else the family contains an uncountable antichain. This dichotomy implies that every nontrivial Souslin poset satisfying the countable chain condition adds a splitting real.We then reduce the dichotomy to a conjecture of Sperner theory. This conjecture concerns the asymptotic behaviour of the product of the sizes of themm-shades of pairs of cross-tt-intersecting families.