A Kruskal–Katona-type theorem for graphs: q-Kneser graphs

Abstract

The “Kruskal-Katona-type problem for a graph G” concerned here is to describe subsets of vertices of G that have minimum number of neighborhoods with respect to their sizes. In this paper, we establish a Kruskal-Katona-type theorem for the q-Kneser graphs, whose vertex set consists of all k-dimensional subspaces of an n-dimensional linear space over a q-element field, two subspaces are adjacent if they have the trivial intersection. It includes as a special case the Erdős–Ko–Rado theorem for intersecting families in finite vector spaces and yields a short proof of the Hilton-Milner theorem for nontrivial intersecting families in finite vector spaces.