**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 $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property:

For every sequence $\langle A_i\mid i<\lambda\rangle$ of unbounded subsets of $\lambda^+$, and every limit $\theta<\lambda$, there exists some $\alpha<\lambda^+$ such that $\text{otp}(C_\alpha)=\theta$, and the $(i+1)_{th}$-element of $C_\alpha$ is a member of $A_i$, for all $i<\theta$.

As an application, we introduce the first construction of an **homogeneous** Souslin tree at the successor of a **singular** cardinal.

In addition, as a by-product, a theorem of Farah and Velickovic (see [FV]), and a theorem of Abraham, Shelah and Solovay (see [AShS:221]) are generalized to cover the case of successors of regulars

**Downloads:**

Pingback: Young Researchers in Set Theory 2011 | Assaf Rinot

Submitted to

Israel Journal of Mathematics, May 2011.Accepted April 2013.

0 likes