On-Line Chain Partitions of Orders

We analyze the on-line chain partitioning problem as a two-person game. One person builds an order one point at a time. The other person responds by making an irrevocable assignment of the new point to a chain of a chain partition. Kierstead gave a strategy showing that width k orders can be on-line chain partitioned into (5 k ?1)=4 chains. We rst prove that width two orders can be partitioned on-line into 5 chains. Secondly, we introduce a variant of the game. We impose the restriction that the new point presented by the rst player has to be a maximum element in the present order. For this up-growing variant we prove matching upper and lower bounds of ? k+1 2 on orders of width k.