Adding a Cohen real adds an entangled linear order

Abstract

The notion of an entangled linear order gives a useful method to construct counter examples for such problems as the productivity of chain conditions, the square bracket partition relation and the existence of a large size monotonic subfunction. In particular, if there exists and ℵ1-entangled linear order then some consequences ofMAℵ1 or of wOCA fail. So, in which model ofZFC does an ℵ1-entangled linear order exist? Todorcevic [6] has shown if cf2ℵ0=ω1 then there is an ℵ1-entangled linear order of size ℵ1. And it is widely known that adding κ many Cohen reals adds such a linear order of size κ. In this note it is shown that adding asingle Cohen real is sufficient to get a maximal size ℵ1-entangled linear order.