Extremal families for Kruskal-Katona Theorem

Abstract

    Given a set of size $n$ and a positive integer $k<n$, Kruskal--Katona theorem gives the minimum size of the shadow of a family $S$ of $k$-sets of $[n]$ in terms of the cardinality of $S$.            We give a characterization of the families of $k$-sets satisfying equality in Kruskal--Katona theorem. This answers a question of F\"uredi and Griggs.