There may be an $n$-entangled set but no $n+1$-entangled sets

Abstract

In this paper we show that for every $2\leq n\in \mathbb{N}$, the statement "there is an $n$-entangled set, but there are no $n+1$-entangled sets" is consistent. We also prove some theorems which improve our understanding of entangled sets in relation to construction schemes: (1) The axiom FCA$^\Delta$ introduced in \cite{finitizationclubch} implies the existence of $n$-entangled sets which are not $n+1$-entangled. (2) $\mathfrak{m}_\mathcal{F}>\omega_1$ implies the non-existence of entangled sets. Thus, $2$-capturing schemes alone are not sufficient to build these kinds of linear orders. (3) The existence of a 2-$\Delta$-capturing scheme is consistent with MA.