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Another Note on Sperner's Lemma

Published online by Cambridge University Press:  20 November 2018

David A. Drake
David A. Drake*
Affiliation:
University of Florida, Gainesville, Florida
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Let Q be a finite partially ordered (by ≤) set with universal bounds O, I. The height function h of Q is defined by the rule: h(x) is the maximum length of a chain from O to x. Let h(I)=n. Suppose that for each k≥0, there exist positive integers a(k) and b(k) such that all elements of height k

  1. (i) are covered by a(k) elements of height k+1;

  2. (ii) cover b(k) elements of height k—1.

Then we call Q a U-poset. Call a subset S of a partially ordered set an antichain if no two elements of S are comparable.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 2 , June 1971 , pp. 255 - 256
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Baker, K. A., A generalization of Sperner's lemma, J. Combinatorial Theory 6 (1969), 224-225.Google Scholar
2. Birkhoff, G., Lattice theory, Colloq. Publ. 25, 3rd ed., Amer. Math. Soc, Providence, R.I., 1967.Google Scholar
3. Lubell, D., A short proof of Sperner's lemma, J. Combinatorial Theory 1 (1966), p. 299.Google Scholar
4. Ore, O., Theory of graphs, Colloq. Publ. 38, Amer. Math. Soc., Providence, R.I., 1962.Google Scholar
5. Sperner, E., Ein Satz über Untermengen einer endlichen Menge, Math. Zeitsch. 27 (1928), 544-548.Google Scholar