Isoperimetric Problems for r-sets

Abstract

How ‘tightly’ can we pack a given number of $r$-sets of an $n$-set? To be a little more precise, let $X=[n]=\{ 1,\ldots,n \}$, and let $X^r=\{ A\subset X : |A|=r \}$. For a set system $\mathcal{A}\subset X^r $, the neighbourhood of $\mathcal{A}$ is $N(\mathcal{A})=\{ B \in X^r: |B \bigtriangleup A|\le 2 \hbox{ for some }A \in \mathcal{A} \}$. In other words, $N(\mathcal{A})$ consists of those $r$-sets that are either in $\mathcal{A}$ or are ‘adjacent’ to it, in the sense that they are at minimal Hamming distance (i.e., distance 2) from some point of it. Given $|\mathcal{A}|$, how small can $|N(\mathcal{A})|$ be?