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Jump number and width. (English) Zbl 0646.06001

Supposing all ordered sets considered in this paper are finite, the authors define the width w(P), the jump number s(P) and a tower number t(P) of an ordered set P. Then they give the proofs that the maximum jump number of ordered sets having width w and tower number t, denoted by s(w,t), satisfies \(c_ 1tw \lg w\leq s(w,t)\leq c_ 2tw \lg w\) for some positive constants \(c_ 1\) and \(c_ 2\). They also give an answer to problem 15 posed by W. T. Trotter [Problems and conjectures in the combinatorial theory of ordered sets, Preprint (1986)], i.e. when w and t are sufficiently large and w is a power of 2, then \((-\epsilon)tw \lg w\leq s(w,t)<(7/10)tw \lg w.\)
Reviewer: R.Firlová

MSC:

06A06 Partial orders, general
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References:

[1] W. Li. The width, tower number, and jump number of order?d sets, to appear in Kexue Tansuo (in Chinese).
[2] W. Li and J. H. Schmerl (1986) Ordered sets with small width and large jump number, Order 3, 1-2. · Zbl 0591.06004
[3] W. T. Trotter (1986) Problems and conjectures in the combinatorial theory of ordered sets, preprint. · Zbl 0664.05001
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