  We prove that in some cases definable thin sets (including chains) of Borel
partial orderings are necessarily countably cofinal. This includes the
following cases: analytic thin sets, ROD thin sets in the Solovay model, and
$\Sigma^1_2$ thin sets in the assumption that $\omega_1^{L[x]}<\omega_1$ for
all reals $x$. We also prove that definable thin wellorderings admit partitions
into definable chains in the Solovay model.
