The study of Q-polynomial distance-regular graphs is an active area of
research.
Many important examples arise on the top fibres of certain meet
semilattices, such as truncated Boolean semilattices and direct products
of "claw semilattices".

In 2003, Brouwer, Godsil, Koolen and Martin introduced the width w and the
dual width w* of a subset in a Q-polynomial distance-regular graph.
They showed that w+w* is at least the diameter D of the graph, and that if
equality holds then the subset has a lot of strong regularities.
For graphs associated with nice semilattices, those subsets satisfying
w+w*=D arise quite naturally within the semilattice structures, and they
are expected to play a role in the classification of Q-polynomial
distance-regular graphs.

On the other hand, recently it is turning out that the subsets with w+w*=D
also play a fundamental role in e.g., coding theory.
In this talk, I will briefly discuss some of the applications of the
theory of these subsets.
Topics include the Erdos-Ko-Rado theorem in extremal set theory, the
Assmus-Mattson theorem in coding and design theory, and orthogonal
polynomials from the Askey scheme.