Joint work with Roy Shalev.
Abstract. We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out de Caux type constructions of topological spaces.
Our main result states that strong instances of $\clubsuit_{AD}$ follow from the existence of a Souslin tree. It is also shown that the weakest instance of $\clubsuit_{AD}$ does not follow from the existence of an almost Souslin tree.
As an application, we obtain a simple, de Caux type proof of Rudin’s result that if there is a Souslin tree, then there is an $S$-space which is Dowker.
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Submitted to Topology and its Applications, October 2020.
Accepted, September 2021.
In an upcoming paper, we answer Question 2.35 in the negative. Furthermore, we prove that the existence of a Luzin set implies $\clubsuit_{AD}(\mathcal S,\omega,{<}\omega)$ for any partition $\mathcal S$ of $\omega_1$ into stationary sets.