Appendix: On Edge–Isoperimetric Theorems for Uniform Hypergraphs

Abstract

Denote by Ω={1,...,n} an n–element set. For all $A,B\in\binom{\Omega}k$, the k–element subsets of Ω, define the relation ~ as follows:

A~B iff A and B have a common shadow, i.e. there is a $C\in\binom{\Omega}{k-1}$ with C ⊂A and C ⊂B. For fixed integer α, our goal is to find a family ${\mathcal A}$ of k–subsets with size α, having as many as possible ~–relations for all pairs of its elements. For k=2 this was achieved by Ahlswede and Katona [2] many years ago.