Jump number and width

AbstractThe maximum jump number of ordered sets having width w and tower number t, denoted by s(w, t), satisfies $$c_l tw\lg w \leqslant s\left( {w,t} \right) \leqslant tw\lg w$$ for some positive constants c1 and c2. Specifically, we can obtain c1=1/8 and c2<11/10. When w and t are sufficiently large and w is a power of 2, then $$\left( {\frac{1}{2} - \varepsilon } \right)tw\lg w \leqslant s\left( {w,t} \right) < \frac{7}{{10}}tw\lg w$$ This gives an answer to a problem posed by W. T. Trotter ([3], Problem 15).