A counterexample to the generalization of Sperner's theorem |
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Authors: | RP DilworthCurtis Greene |
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Institution: | California Institute of Technology, Pasadena, California 91109 USA |
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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. |
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