  This paper studies pliable index coding, in which a sender broadcasts
information to multiple receivers through a shared broadcast medium, and the
receivers each have some message a priori and want any message they do not
have. An approach, based on receivers that are absent from the problem, was
previously proposed to find lower bounds on the optimal broadcast rate. In this
paper, we introduce new techniques to obtained better lower bounds, and derive
the optimal broadcast rates for new classes of the problems, including all
problems with up to four absent receivers.
