Isoperimetric Inequalities for Faces of the Cube and the Grid

Abstract

A face of the cube ℘( N ) = {0,1} N is a subset determined by fixing the values of some coordinates and allowing the remainder free rein. For instance, the edges of the cube are faces of dimension 1. In Section 2 of this paper we prove a best possible upper bound for the number of i -faces of ℘( N ) contained in any subset of ℘( N ). In particular, we show that initial segments in the binary ordering the ordering on ℘( N ) induced by the map A ↦ Σ i ∈ A 2 i : ℘( N )→ ℕ—contain the greatest possible number of i -faces for any i ⩾0. In Section 3 the inequality is extended to apply to the grid [ p ] N for p ⩾ 2, and to give a bound on the number of i -dimensional faces enclosed by a collection of j -dimensional faces, for i ⩾ j . Finally, in Section 4, we apply the face isoperimetric result to the problem which originally motivated its study. We prove a Kruskal-Katona type result for down-sets in the grid.