The Ostaszewski square, and homogenous Souslin trees

TL;DR

This paper proves a new guessing property for the square principle under GCH, and uses it to construct homogeneous Souslin trees at successors of singular cardinals, extending previous theorems to new cases.

Contribution

It introduces a remarkable guessing property for the square principle and constructs homogeneous Souslin trees at successors of singular cardinals under GCH.

Findings

Established a new guessing property for _ under GCH.

Constructed homogeneous  trees at successors of singular cardinals.

Generalized existing theorems to cover successors of regular cardinals.

Abstract

Assume GCH and let $\lambda$ denote an uncountable cardinal. We prove that if $\square_\lambda$ holds, then this may be witnessed by a coherent sequence $< C_\alpha | \alpha < \lambda^+ >$ with the following remarkable guessing property: For every sequence $< A_i | i<\lambda >$ of unbounded subsets of $\lambda^+$, and every limit $\theta<\lambda$, there exists some $\alpha<\lambda^+$ such that $\otp(C_\alpha)=\theta$, and the $(i+1)_{th}$-element of $C_\alpha$ is a member of $A_i$, for all $i<\theta$. As an application, we construct an homogenous $\lambda^+$-Souslin tree from $GCH+\square_\lambda$, for every singular cardinal $\lambda$. In addition, as a by-product, a theorem of Farah and Velickovic, and a theorem of Abraham, Shelah and Solovay are generalized to cover the case of successors of regulars.