Jump number and width (Q1103654)

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 \textit{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 \((-\epsilon)tw \lg w\leq s(w,t)<(7/10)tw \lg w.\)