Irreducible Width 2 Posets of Linear Discrepancy 3

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |h _L (x) − h _L (y)| ≤ k, where h _L (x) is the height of x in L. Tannenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem for characterizing the posets of linear discrepancy 2. Howard et al. (Order 24:139–153, 2007) showed that this problem is equivalent to finding all posets of linear discrepancy 3 such that the removal of any point reduces the linear discrepancy. In this paper we determine all of these minimal posets of linear discrepancy 3 that have width 2. We do so by showing that, when removing a specific maximal point in a minimal linear discrepancy 3 poset, there is a unique linear extension that witnesses linear discrepancy 2. The first author was supported during this research by National Science foundation VIGRE grant DMS-0135290.