Results on Linear Discrepancy

The linear discrepancy of a poset P , denoted ld(P ), is the minimum, over all linear extensions L, of the maximum distance in L between two elements incomparable in P . We prove that ld(P ) ≤ ⌊(3r − 1)/2⌋ when P has width 2. Tanenbaum, Trenk, and Fishburn asked whether this upper bound holds for all posets. We answer this in the negative by giving a randomized construction of bipartite posets whose linear discrepancy is asymptotic to the trivial upper bound 2r − 1. For products of chains, we give alternative proofs of results that have been obtained independently elsewhere.