Linear ROD subsets of Borel partial orders are countably cofinal in Solovay's model

Abstract

The following is true in the Solovay model. 1. If $\leq$ is a Borel partial quasi-order on a Borel set $D$ of the reals, $X$ is a ROD subset of $D$, and $\leq$ restricted to $X$ is linear, then $X$ is countably cofinal in the sense of $\leq$. 2. If in addition every countable set $Y$ of $D$ has a strict upper bound in the sense of $\leq$, then the ordering $< D ; \leq >$ has no maximal chains that are ROD sets.