A Bound on the Dimension of Interval Orders

We make use of the partially ordered set (I(0, n), <) consisting of all closed intervals of real numbers with integer endpoints (including the degenerate intervals with the same right- and left-hand endpoints), ordered by [a, b] < [c, d] if b < c, to show that there is no bound on the order dimension of interval orders. We then turn to the problem of computing the dimension of I(0, n), showing that I(0, 10) has dimension 3 but I(0, 11) has dimension 4. We use these results as initial conditions in obtaining an upper bound on the dimension of I(0, n) as a logarithmic function of n. It is our belief that this example is a “canonical” example for interval orders, so that the computation of its dimension should have significant impact on the problem of computing the dimension of interval orders in general.