An easy subexponential bound for online chain partitioning
  Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width $w$ into $w^{14\lg w}$ chains. We improve this to $w^{6.5 \lg w + 7}$ with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach. 
Bart{\l}omiej Bosek, Hal A. Kierstead, Tomasz Krawczyk, Grzegorz   Matecki, Matthew E. Smith
cs.DS math.CO