The Ostaszewski square and homogeneous Souslin trees

Assume GCH and let lambda denote an uncountable cardinal. We prove that if a-(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 a homogeneous lambda(+)-Souslin tree from a-(lambda) + CH lambda, for every singular cardinal lambda. In addition, as a by-product, a theorem of Farah and VelikoviA double dagger, and a theorem of Abraham, Shelah and Solovay are generalized to cover the case of successors of regulars.