<p>In the late 1960's, Kruskal and Katona solved independently an isoperimetric problem in the high-dimensional simplex. A general Kruskal-Katona-type problem on graphs is to describe subsets of the vertex set of a graph with minimum number of neighborhoods with respect to its their own sizes. We reort a few of Kruskal-Katona-type theorems for graphs, especially for the derangement graph of the symmetric group on a finite set. With this theorem we deduce the size and structure of the first three maximal intersecting families in the symmetric group, where the first was given by Deza-Frankl and Cameron-Ku; the second was conjectured by Cameron-Ku. With this theorem we also determine the maximum product of two cross-intersecting families in the symmetric group under various conditions.</p>
