  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.
