According to Efficient algorithms for interval graphs and circular arc graphs there is an $O(n \log n)$ algorithm for finding the max clique in an interval graph, assuming you have the interval model. Unfortunately, neither the paper nor the references actually elaborate on what the algorithm is, beyond saying that you can modify the well-known algorithm for optimally coloring interval graphs (where you greedily color the graph in the order generated by sorting the intervals lexicographically) to compute the max clique. This makes sense, as the largest color used in an optimal coloring is the size of the max clique in a perfect graph (and interval graphs are perfect). Unfortunately, beyond that, I am lost as to how to compute the max clique. The obvious approach is to modify some data structure every time the max color increases, but I can't figure out precisely what data structure to use or how to modify it.
Does anyone know a reference for this algorithm or can reinvent a description of it?