SOME QUESTIONS ON ENTANGLED LINEAR ORDERS

Entangled linear orders were first introduced by Abraham and Shelah. Todorčević showed that these linear orders exist under $\mathsf{CH}$. We prove the following results: (1) If $\mathsf{CH}$ holds, then, for every $n > 0$, 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 $\mathbb{R}\subseteq \mathrm{L}$, then there is an entangled $Π_1^1$ set of reals. (4) If $\diamondsuit$ holds, then there is a $2$-entangled non-separable linear order.