The Ostaszewski square, and homogenous Souslin trees

Assume GCH and let $λ$ denote an uncountable cardinal. We prove that if $\square_λ$ holds, then this may be witnessed by a coherent sequence $< C_α| α< λ^+ >$ with the following remarkable guessing property: For every sequence $< A_i | i<λ>$ of unbounded subsets of $λ^+$, and every limit $θ<λ$, there exists some $α<λ^+$ such that $\otp(C_α)=θ$, and the $(i+1)_{th}$-element of $C_α$ is a member of $A_i$, for all $i<θ$. As an application, we construct an homogenous $λ^+$-Souslin tree from $GCH+\square_λ$, for every singular cardinal $λ$. 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.

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