On the widths of finite distributive lattices

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Abstract

The following conjecture of U Faigle and B Sands is proved: For every number R > 0 there exists a number n(R) such that if L is a finite distributive lattice whose width w(L) (size of the largest antichain) is at least n(R), then |L| ⩾ Rw(L). In words this says that as one considers increasingly large distributive lattices, the maximum sized antichain contains a vanishingly small proportion of the elements.

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Supported in part by NSF Grant MCS83-01867.

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Supported in part by a Sloan Research Fellowship.