No abstract

No abstract

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P ) ≤ d. We answer this question in the affirmative by showing that d = 17 is sufficient. We also show that if P is a poset containing the standard example S 5 as a subposet, then the cover graph of P has treewidth at least 3.

In this paper, we answer a question posed by Y.H. Shen. We prove that if I is an m-generated squarefree monomial ideal in the polynomial ring S = K [x 1 , . . . , x n ] with K a field, then sdepth I n − m/2 . The proof is inductive and uses the correspondence between a Stanley decomposition of a monomial ideal and a partition of a particular poset into intervals established by Herzog, Vladoiu and Zheng.

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,Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing the posets of linear discrepancy 2. We show that this problem is equivalent to finding the posets with linear discrepancy equal to 3 having the property that the deletion of any point results in a reduction in the linear discrepancy. Howard determined that there are infinitely many such posets of width 2. We complete the forbidden subposet characterization of posets with linear discrepancy equal to 2 by finding the minimal posets of width 3 with linear discrepancy equal to 3. We do so by showing that, with a small number of exceptions, they can all be derived from the list for width 2 by the removal of specific comparisons.