Uniform asymptotics for the Kleitman-West problem

Abstract

Two distinct $r$-element sets are adjacent if their intersection has size $r-1$. The Kleitman-West problem, equivalently formulated as the edge-isoperimetric problem on the Johnson graph $J(n,r)$, asks for the maximum number $M_r(n,m)$ of unordered adjacent pairs in a family of $m$ $r$-element subsets of an $n$-element set, and for the families attaining this maximum. For every fixed integer $r\ge2$, we determine this maximum asymptotically as $n\to\infty$, uniformly over the entire range of family sizes. Specifically, for each integer $1\le j\le r$, define $F_j(p)=\max\{p^{1+1/j},\,2p-1+(1-p)^{1+1/j}\}$ for $0\le p\le1$, and $G_j(x)=\lfloor x\rfloor+F_j(x-\lfloor x\rfloor)$ for $x\ge0$. With $P_r(n,m)=\sum_{j=1}^r\frac{n^{j+1}}{2(j-1)!}G_j\!\left(\frac{j!m}{n^j}\right)-\frac{r(r-1)}4mn-\frac m2,$ we prove that $\lim_{n\to\infty}\max_{2\le m\le\binom nr}\left|\frac{M_r(n,m)}{P_r(n,m)}-1\right|=0.$ The remaining cases satisfy $M_r(n,0)=M_r(n,1)=0$. The proof first establishes the asymptotic formula for dense families using a sharp inequality for bounded random variables, with equality attained by a random variable taking at most two values. To extend the result uniformly to all family sizes, we extract groups of sets sharing a fixed subset, apply the dense estimates after deleting that subset, and control the adjacent pairs not contained in any one group.