  We study the computational complexity of the FO model checking problem on
interval graphs, i.e., intersection graphs of intervals on the real line. The
main positive result is that FO model checking and successor-invariant FO model
checking can be solved in time O(n log n) for n-vertex interval graphs with
representations containing only intervals with lengths from a prescribed finite
set. We complement this result by showing that the same is not true if the
lengths are restricted to any set that is dense in an open subset, e.g., in the
set $(1, 1 + \varepsilon)$.
