The Sub-exponential Upper Bound for On-Line Chain Partitioning

The main question in the on-line chain partitioning problem is to determine whether there exists an algorithm that partitions on-line posets of width at most $w$ into polynomial number of chains – see Trotter's chapter Partially ordered sets in the Handbook of Combinatorics. So far the best known on-line algorithm of Kier stead used at most $(5^w-1)/4$ chains, on the other hand Szemer\'{e}di proved that any on-line algorithm requires at least $\binom{w+1}{2}$ chains. These results were obtained in the early eighties and since then no progress in the general case has been done. We provide an on-line algorithm that partitions orders of width $w$ into at most $w^{16\log{w}}$ chains. This yields the first sub-exponential upper bound for on-line chain partitioning problem.