On the height of a random set of points in a d-dimensional unit cube

We investigate, through numerical experiments, the asymptotic behavior of the length H-d(n) of a maximal chain (longest totally ordered subset) of a set of n points drawn from a uniform distribution on the d-dimensional unit cube V-D = [0, 1](d). For d greater than or equal to 2, it is known that c(d)(n) = H-d(n)/n(1/d) converges in probability to a constant c(d) < e, with lim(d-->infinity) c(d) = e. For d = 2, the problem has been extensively studied, and it is known that c(2) = 2; c(d) is not currently known for any d greater than or equal to 3. Straightforward Monte Carlo simulations to obtain c(d) have already been proposed, and shown to be beyond the scope of current computational resources. In this paper, we present a computational approach which yields feasible experiments that lead to estimates for c(d). We prove that H-d(n) can be estimated by considering only those chains close to the diagonal of the cube. A new conjecture regarding the asymptotic behavior of c(d)(n) leads to even more efficient experiments. We present experimental support for our conjecture, and the new estimates of c(d) obtained from our experiments, for d is an element of {3, 4, 5, 6}.