Chain Partitions

In the text, a proof is given of a result with a similar statement that a partially ordered set with height k is a union of k C 1 antichains (the number is k C 1 because the height of partially ordered set is the defined to be the number of links in the longest chain, and there is one more node in the chain than the number of links). The introduction to Dilworth’s Theorem claims that it has a similar proof. This claim is misleading. Most student attempts at a proof sought a natural chain whose removal would decrease the width, but were incomplete. Of course, the theorem says that such chains must exist, but there does not seem to be an easy way to find one directly. A standard approach is known as the greedy method: identify the largest object of the desired type and remove it to leave a simpler problem. One expects that this would apply here, but a proof that the width is decreased is elusive. Moreover, the cases in which it is easiest to show that the width is decreased involve the removal of a short chain. The case of an isolated point, incomparable to all others, is an extreme example. This suggests that a different method will be required. We will retutn to this later.