A containment result in ${\mathbb{P}}^n$ and the Chudnovsky conjecture

In the paper we prove the containment $I^{(nm)}\subset M^{(n-1)m}{I}^m$, for a radical ideal $I$ of $s$ general points in ${\mathbb{P}}^n$, where $s\ge 2^n$. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least $2^n$ points in ${\mathbb{P}}^n$.