Nonhomogeneous analytic families of trees

TL;DR

This paper establishes a dichotomy for analytic families of trees, linking coloring properties and antichain structures, and connects these results to a conjecture in Sperner Theory about intersecting families.

Contribution

It introduces a new dichotomy for analytic families of trees and relates it to a Sperner Theory conjecture, advancing understanding of tree structures and their combinatorial properties.

Findings

Either all but finitely many levels are nonhomogeneous under some coloring or the family contains an uncountable antichain.

Every nontrivial Souslin poset satisfying ccc adds a splitting real.

Reduces the dichotomy to a conjecture about the asymptotic behavior of cross-t-intersecting families.

Abstract

We consider a dichotomy for analytic families of trees stating that either there is a colouring of the nodes for which all but finitely many levels of every tree 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 is concerning the asymptotic behaviour of the product of the sizes of the m-shades of pairs of cross-t-intersecting families.