  A famous theorem of Dilworth asserts that any finite poset of width $k$ can
be decomposed into $k$ chains. We study the following problem: given a Borel
poset $P$ of finite width $k$, is it true that it can be decomposed into $k$
Borel chains? We give a positive answer in a special case of Borel posets
embeddable into the real line. We also prove a dual theorem for posets whose
comparability graphs are locally countable.
