Abstract
It has been conjectured that the analog of Sperner’s theorem on non-comparable subsets of a set holds for arbitrary geometric lattices, namely, that the maximal number of non-comparable elements in a finite geometric lattice is max w(k), where w(k) is the number of elements of rank k. It is shown in this note that the conjecture is not true in general. A class of geometric lattices, each of which is a bond lattice of a finite graph, is constructed in which the conjecture fails to hold.
This research was partially supported by NSF Grant GP 8423
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References
E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Math. Z. 27 (1928), 544–548.
L.H. Harper, The Morphology of Geometric Lattices (to appear).
K.A. Baker, A Generalization of Sperner’s Lemma (to appear).
J.E. McLaughlin, Structure Theorems for Relatively Complemented Lattices, Pacific J. Math. 3 (1953), 197–208.
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Dilworth, R.P., Greene, C. (1990). A Counterexample to the Generalization of Sperner’s Theorem. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_27
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DOI: https://doi.org/10.1007/978-1-4899-3558-8_27
Publisher Name: Birkhäuser, Boston, MA
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