Some questions on entangled linear orders

TL;DR

This paper investigates properties and existence results of entangled linear orders under various set-theoretic assumptions, extending previous work by Abraham, Shelah, and Todorčević.

Contribution

It proves new existence results for n-entangled linear orders and entangled sets of reals under assumptions like CH, L, and iamondsuit.

Findings

Existence of n-entangled linear orders not (n+1)-entangled under CH.

Existence of two homeomorphic sets of reals with different entanglement properties under CH.

Existence of entangled  sets of reals if  is in L.

Abstract

Entangled linear orders were first introduced by Abraham and Shelah. Todor\v{c}evi\'c 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 $\Pi_1^1$ set of reals. (4) If $\diamondsuit$ holds, then there is a $2$-entangled non-separable linear order.