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An extension of stone's partitioning theorem for ordered sets

Part of: Combinatorics

Published online by Cambridge University Press:  26 February 2010

J. W. Taylor
J. W. Taylor
Affiliation:
Department of Mathematics, University of Illinois, Urbana.
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Extract

In [1], A. H. Stone proved that for a cardinal number k ≥ 1 a set with a transitive relation can be partitioned into k cofinal subsets provided each element of the set has at least k successors. Using methods quite different from those of Stone, we show that for k ≥ ℵ0 the same condition on successors guarantees that a set on which there are defined not more than k transitive relations can be partitioned into k sets each of which is cofinal with respect to each of the relations. We also show that such a partition exists even if some of the relations are not transitive as long as the non-transitive relations have no more than k elements.

MSC classification

Secondary: 05A17: Partitions of integers
Type
Research Article
Information
Mathematika , Volume 18 , Issue 1 , June 1971 , pp. 5 - 7
Copyright
Copyright © University College London 1971

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References

1.Stone, A. H., “On partitioning ordered sets into cofinal subsets”, Mathematika, 15 (1968), 217–222.CrossRefGoogle Scholar