Abstract: Given a linear extension of a poset P, let h be the function giving the height of each element on the extension. The linear discrepancy ld(P) is the least integer m for which there exists a linear extension such that |h(x)-h(y)| <= m if x and y are incomparable. The exact value of the linear discrepancy of a product of two chains is known; In 2003 Hong, Hyun, and Kim proved that the linear discrepancy of the product of n-element and m-element chains is the ceiling of (1/2)mn-2.
In this talk we present asymptotic bounds for the linear discrepancy of the product of three k-chains and the linear discrepancy of the product of four k-chains. In three dimensions, the value is (3/4+o(1))k3. In four dimensions, the value is (7/8+o(1))k4. The upper bound construction generalizes easily to d dimensions. This is joint work with Douglas B. West.