Abstract
The classical theorem of R. P. Dilworth asserts that a partially ordered set of width n can be partitioned into n chains. Dilworth's theorem plays a central role in the dimension theory of partially ordered sets since chain partitions can be used to provide embeddings of partially ordered sets in the Cartesian product of chains. In particular, the dimension of a partially-ordered set never exceeds its width. In this paper, we consider analogous problems in the setting of recursive combinatorics where it is required that the partially ordered set and any associated partition or embedding be described by recursive functions. We establish several theorems providing upper bounds on the recursive dimension of a partially ordered set in terms of its width. The proofs are highly combinatorial in nature and involve a detailed analysis of a 2-person game in which one person builds a partially ordered set one point at a time and the other builds the partition or embedding.
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References
D.Bean (1976) Effective coloration, J. Symbolic Logic 41, 469–480.
D.Bean (1976) Recursive Euler and Hamilton paths, Proc. Amer. Math. Soc. 55, 385–394.
K.Bogart, I.Rabinovitch, and W.Trotter (1976) A bound on the dimension of interval orders, J. Comb. Theory 21, 319–328.
R. P.Dilworth (1950) A decomposition theorem for partially ordered sets, Ann. of Math. 51, 161–166.
B.Dushnik and E.Miller (1941) Partially ordered sets, Amer. J. Math, 63, 600–610.
L. Hopkins (1981) Some problems involving combinational structures determined by intersections of intervals and arcs, PhD Thesis, University of South Carolina, pp. 1–91.
H.Kierstead (1981) An effective version of Dilworth's theorem, Trans. Amer. Math. Soc. 268, 63–77.
H.Kierstead (1981) Recursive colorings of highly recursive graphs, Canad. J. Math. 33, 1279–1290.
H.Kierstead (1983) An effective version of Hall's theorem, Proc. Amer. Math. Soc. 88, 124–128.
H.Kierstead and W.Trotter (1981) An extremal problem in recursive combinatorics, Congressus Numerantium 33, 143–153.
A.Manaster and J.Remmel (1981) Partial orderings of fixed finite dimension: model companions and density, J. Symbolic Logic 46, 789–802.
A.Manaster and J.Rosenstein (1980) Two-dimensional partial orderings: recursive model theory, J. Symbolic Logic 45, 121–143.
G.McNulty (1982) Infinite ordered sets, a recursive perspective, Proceedings of the Symposium on Ordered Sets (I.Rival ed.), Reidel, Dordrecht, 1982.
O.Ore (1962) Theory of Graphs, Colloquium Publication 38, Amer. Math. Soc. Providence, RI 1962.
J. Remmel (1983) On the effectiveness of the Schröder-Bernstein Theorem, Proc. Amer. Math. Soc.
H.Rogers (1967) Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York.
J.Rosenstein (1982) Linear Orders, Academic Press, New York.
J.Schmerl (1980) Recursive colorings of graphs, Canad. J. Math. 32, 821–830.
J.Schmerl (1982) The effective version of Brooks theorem, Canad. J. Math. 34, 1036–1046.
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Communicated by D. Kelly
This paper was prepared while the authors were supported, in part, by NSF grant ISP-80-11451. In addition, the second author received support under NSF grant MCS-80-01778 and the third author received support under NSF grant MCS-82-02172.
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Kierstead, H.A., McNulty, G.F. & Trotter, W.T. A theory of recursive dimension for ordered sets. Order 1, 67–82 (1984). https://doi.org/10.1007/BF00396274
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DOI: https://doi.org/10.1007/BF00396274