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Let P be a poset in which each point is incomparable to at most ∆ others. Tanen-baum, Trenk, and Fishburn asked in a 2001 paper if the linear discrepancy of such a poset is bounded above by (3∆ − 1)/2. In this paper, we answer this question in the affirmative for two classes of posets by proving upper bounds in terms of ∆. We first prove a Brooks-type bound… (More)

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. Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing… (More)

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. 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… (More)

a r t i c l e i n f o a b s t r a c t In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1, 2,. .. ,n} into intervals. Specifically,… (More)

th birthday. Among his many remarkable contributions to combinatorial mathematics and theoretical computer science is a jewel for online problems for partially ordered sets: the fact that h(h + 1)/2 antichains are required for an online antichain partition of a poset of height h. The linear discrepancy of a poset P is the least k for which there is a linear… (More)

Let K be a field and S = K[x 1 ,. .. , x n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth(M), and conjectured that depth(M) ≤ sdepth(M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case… (More)

To my parents, Jim and Karol, who believed a farm kid from North Dakota could do whatever he wanted and helped me achieve whatever I imagined and more. iii ACKNOWLEDGEMENTS There are many people without whose mentoring, guidance, support, and friendship this dissertation never would have been written. My advisor, William T. Trotter, is at the top of that… (More)

The Füredi–Hajnal conjecture, originally posed in [6] and solved by Marcus and Tardos in [10], can be cast as an extremal problem on ordered graphs. It states that any n-vertex bipartite graph that avoids a fixed permutation graph has O(n) edges. Balogh, Bollobás and Morris [1] recently generalized this result to show that any n-vertex, ordered hypergraph… (More)

This honors thesis introduces some fundamental ideas of knot theory in a way that is accessible to nonmathematicians. It summarizes some of the major historical developments in the mathematical theory of knots, beginning with Thomson and Tait and ending with some of the important results of the late 20 th century. A few important examples of ways in which… (More)