Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs

Brouwer, Godsil, Koolen and Martin [Width and dual width of subsets in polynomial association schemes, J. Combin. Theory Ser. A 102 (2003) 255-271] introduced the width w and the dual width w^* of a subset in a distance-regular graph and in a cometric association scheme, respectively, and then derived lower bounds on these new parameters. For instance, subsets with the property w+w^*=d in a cometric distance-regular graph with diameter d attain these bounds. In this paper, we classify subsets with this property in Grassmann graphs, bilinear forms graphs and dual polar graphs. We use this information to establish the Erd?s-Ko-Rado theorem in full generality for the first two families of graphs.