Various relations for comparison of intervals of real numbers are introduced, and the expected length of the corresponding longest increasing subsequence is analyzed. When intervals are randomly generated by taking the minimum and maximum of two independent uniform random variables, we prove that the expected length of the longest increasing subsequence grows on root the order of 3 n. We also investigate the asymptotic behavior of the expected length under alternative comparison relations and random interval models. Discussions on other subsequence problems for interval sequences are included.
