Independent sets in every large subset of a dense random graph ($|U|\ge n/(\log n{)}^2⟹\alpha (G[U])\ge (2-\gamma ){\log }_{1/(1-p)}n$)

For fixed $p$ and $\gamma >0$, Janson inequality bounds the failure probability for a given large vertex subset by $\exp (-c|U{|}^2/(\log n{)}^4)$. A union bound over all at most $2^n$ subsets still tends to zero. This uniform statement is essential because vertex sets left by a greedy colouring procedure depend on the graph; one cannot assume each adaptively chosen remainder is an independent fresh random graph.