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 $\otp(C_\alpha)=\theta$, 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.