  We present a refinement of the classical alteration method for constructing
$H$-free graphs: for suitable edge-probabilities $p$, we show that removing all
edges in $H$-copies of the binomial random graph $G_{n,p}$ does not
significantly change the independence number. This differs from earlier
alteration approaches of Erd\H{o}s and Krivelevich, who obtained similar
guarantees by removing one edge from each $H$-copy (instead of all of them). We
demonstrate the usefulness of our refined alternation method via two
applications to online graph Ramsey games, where it enables easier analysis.
