A note on ranking functions

Abstract

In this issue, W.J. Walker introduces the lattice L(n, r) as the set of all possible results when n competitors are matched in a series of r races. A result is an r-term nondecreasing sequence of integers selected from {1,2,…, n}. The dimension of L(n, r) is at most r since it is a subposet of R t . Walker conjectures that L(n, r) is at fact the intersection of r consistent linear extensions and verifies this conjecture when r ⩽ 2 as well as for the case ( n, r) = (4, 3). In this note, we show that the general conjecture does not hold by providing that for every r >/ 3 and every t ⩾ r, there exists an integer n 0 so that if n ⩾ n 0, then L(n, r) is not the intersection of t consistent linear extensions.