Sharp Bounds on the Number of Small Cuts

Let $λ$ be the minimum cut value of an $n$-vertex undirected multigraph. For every fixed $α>1$, we prove that there are $O(n^{\lceil2α\rceil-1})$ cuts of size strictly below $αλ$. The exponent is sharp. The proof combines splitting off and sampling with a bound on the size of nested families of vertex sets.