We give a proof of the Stanley-Stembridge conjecture on chromatic symmetric functions for the class of all unit interval graphs with independence number 3. That is, we show that the chromatic symmetric function of the incomparability graph of a unit interval order in which the length of a chain is at most 3 is positively expanded as a linear sum of elementary symmetric functions.
The authors are grateful to the referee for careful reading of the paper and suggestions, which let us improve the clarity of the paper. The most of the work on this paper was done while the first named author was visiting Korea Institute for Advance Study(KIAS) and the second named author was working there. The authors are grateful to KIAS for the hospitality.
