  An on-line chain partitioning algorithm receives the elements of a poset one
at a time, and when an element is received, irrevocably assigns it to one of
the chains. In this paper, we present an on-line algorithm that partitions
posets of width $w$ into $w^{O(\log{\log{w}})}$ chains. This improves over
previously best known algorithms using $w^{O(\log{w})}$ chains by Bosek and
Krawczyk and by Bosek, Kierstead, Krawczyk, Matecki, and Smith. Our algorithm
runs in $w^{O(\sqrt{w})}n$ time, where $w$ is the width and $n$ is the size of
a presented poset.
