SOME QUESTIONS ON ENTANGLED LINEAR ORDERS

Abstract Entangled linear orders were introduced by Abraham and Shelah in [2]. Todorčević [29] showed that these linear orders exist under $\mathsf {CH}$ . We prove the following results: (1) If $\mathsf {CH}$ holds, then, for every $n> 1$ , there is an n -entangled linear order which is not $(n+1)$ -entangled. (2) If $\mathsf {CH}$ holds, then there are two homeomorphic sets of reals $A,B \subseteq \mathbb {R}$ such that A is entangled but B is not $2$ -entangled. (3) If $\mathbb {R} \subseteq \mathsf {L}$ , then there is an entangled $\Pi _1^1$ set of reals. (4) If $\diamondsuit $ holds, then there is a $2$ -entangled non-separable linear order.