The Ostaszewski square, and homogenous Souslin trees

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 $\otp(C_α)=θ$, 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.

preliminary version