By R. P. Dilworth (auth.), Kenneth P. Bogart, Ralph Freese, Joseph P. S. Kung (eds.)

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We shall give one example here. An important special case of Dilworth's theorem is the case when the ordered set P is constructed from a finite sequence (al' a2, ... , an) of positive integers in the following way: P is the set of ordered pairs (ai, i), ordered by the prod uct order: (ai, i) < (aj, j) if ai ~ aj and i < j. In P, Chain Partitions in Ordered Sets 19 a chain is a nondecreasing subsequence and an antichain is a nonincreasing subsequence. From this, one can derive the following theorem of Erdos and Szekeres [16] by a simple counting argument: Any finite sequence of length n 2 + 1 contains a monotone subsequence of length n + 1.

With the Erdos-Szekeres theorem as a model, Dilworth's theorem can be recast as follows: Any ordered set P of size at least ab + 1 contains either a chain of length of a + 1 or an antichain of size b + 1. The lower bound ab + 1 is best possible and thus, Dilworth's theorem allows the calculation of an exact Ramsey function. 2. Alternative proofs. Since 1950, many alternative proofs of Dilworth's decomposition theorem have appeared. These proofs can be divided roughly into two groups: direct elementary arguments and indirect arguments deriving Dilworth's theorem from other theorems in combinatorics.

Deming [13] showed that for G an undirected graph, f3o( G) equals the maximum over all acyclic orientations w of the size of a minimum chain decomposition of the directed graph G oriented according to w. Dilworth's theorem follows from this result. Finally, Dilworth's theorem is equivalent to the theorem that the incomparability graph of an ordered set (that is, the graph on the elements of the ordered set with two elements joined by an edge whenever they are incomparable) is perfect (that is, its chromatic number equals the size of a maximum-sized clique).