Nonhomogeneous analytic families of trees

2008/07/01 by James Hirschorn, Hirschorn, James · 1 citation
Mathematics · #03E15 (Primary) 03E40 #05D05 #28A12 (Secondary) #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.0807.0147

openalex publication_date 2008/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Asymptotic upper bounds on the shades of t-intersecting families

Related

• Set families with a forbidden induced subposet
• Hechler and Laver Trees
• The vanishing levels of a tree
• Set families with a forbidden subposet
• The Ostaszewski square, and homogenous Souslin trees