An effective version of Dilworth’s theorem

Abstract

0. Introduction. Loosely speaking, a subset A of the natural numbers is recursive iff there exists an algorithm (i.e., a finite computer program) which upon input of a natural number n outputs if n E A and 0 otherwise. Similarly a partial ordering (A, A) is recursive iff there is an algorithm which upon input of an ordered pair of natural numbers (a, b) outputs 1 if a < b and 0 otherwise. For a more careful definition of recursive relations see [R]. One of the attractions of linite combinatorics over infinite combinatorics is its explicit constructions. One never has to consider whether a finite object really exists. This paper is part of a program to enlarge the domain of finite combinatorics to certain infinite structures while preserving the explicit constructions of the smaller domain. We shall consider a recursive combinatorics whose domain is the recursive structures. Since finite structures are trivially recursive this domain does indeed extend that of finite combinatorics. Moreover each structure in this domain is explicitly exhibited by some finite computer program. Questions from the graph theory of this combinatorics have been studied by Bean [B], [Bi], Kierstead [K], and Schmerl [S], [Si]. Generally their results relate the chromatic number of a recursive graph with certain properties to its recursive chromatic number. The following example illustrates these ideas. Dilworth's theorem [D] asserts that any partial ordering of finite width n can be covered by n chains. If the partial ordering is finite, then one can actually exhibit these chains (by trial and error, if by no other method). The following easy argument demonstrates how Dilworth's theorem for countably infinite partial orderings follows from Dilworth's theorem for finite partial orderings. Let P = {pi: i E N). We show by induction that for all i E N there exist chains CO,..., C,:'-1 such that: (i) ifj < i and k < n then Ck C k (ii)pi E CO+l U ... U Cn_1; (iii) if Q is a finite subset of P then Co, . . ., C,'1 can be extended to chains that cover Q.