Entangled Linear Orders in the Easton's Models

Abstract

The notion of an entangled linear order was first introduced by Avraham and Shelah [1]. Subsequently, Todorcevic [5] generalized it to higher cardinals and mentioned it is useful to solve problems such as the productivity of chain conditions and the square bracket partition relations. He also showed that if $\mathbf{wCH}(\mu)$ holds there is a $2^{\mu}$-entangled linear order of size $2^{\mu}$. From this, we can immediately observe that $\mathbf{GCH}$ implies the full existence of entangled linear orders. In this paper we will show that such full existence occurs also in the Easton's models in which we can arbitrarily determine the powers of infinite regular cardinals.