  We prove that the family of largest cuts in the binomial random graph
exhibits the following stability property: If $1/n \ll p = 1-\Omega(1)$, then,
with high probability, there is a set of $n - o(n)$ vertices that is
partitioned in the same manner by all maximum cuts of $G_{n,p}$. Moreover, the
analogous statement remains true when one replaces maximum cuts with
nearly-maximum cuts.
  We then demonstrate how one can use this statement as a tool for showing that
certain properties of $G_{n,p}$ that hold in a fixed balanced cut hold
simultaneously in all maximum cuts. We provide two example applications of this
tool. First, we prove that maximum cuts in $G_{n,p}$ typically partition the
neighbourhood of every vertex into nearly equal parts; this resolves a
conjecture of DeMarco and Kahn for all but a narrow range of densities $p$.
Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs
$H$, we prove a lower bound on the threshold density $p$ above which every
largest $H$-free subgraph of $G_{n,p}$ is $(\chi(H)-1)$-partite. Our lower
bound exactly matches the upper bound on this threshold recently obtained by
the first two authors.
