Let $P_1,\cdots,P_m$ be points in $\mathbb{R}^n$ such that the distance between any $P_i,P_j$ is less than 1.
Let $p_{i,j}$ be a probability distribution on pairs of these points, that is for $1\leq i,j\leq m$, $p_{i,j}\geq 0$ and $\sum_{i,j} p_{i,j}=1$.
Let $\epsilon>0$ be such that $$\sum_{d(P_i,P_j)>\epsilon} p_{i,j}<\epsilon.$$
Is there a way to choose $\epsilon'$ and $\delta$ and cluster these points into groups satisfying these three conditions?
$\epsilon'$ and $\delta$ only depend on $\epsilon$, not on $n,m$. As $\epsilon$ goes to 0, $\epsilon'$ and $\delta$ go to 0 also.
Any two points in the same group have distance less than $\epsilon'$.
The total probability of pairs of points from different groups is less than $\delta$.
