= Continuity of fixed-choice percolation
A fixed-choice <Achlioptas process> cannot have its <largest component of a graph> jump from $o(n)$ to a fixed positive fraction of $n$ within $o(n)$ steps, <with high probability>. The proof combines <forced merging of large components> and <persistence of unsampled graph components>. An assumed jump forces positive vertex mass in each band $[k,Dk)$ slightly before the jump: insufficient smaller components can feed the final large component, while enough components above $Dk$ would merge too early. Persistence carries a fixed positive amount of each band to a common time. Taking sufficiently many disjoint bands with $k=1,D,D^2,\ldots$ then counts more than $n$ <vertices>. The number of offered candidates is fixed throughout this argument. Riordan and Warnke establish this obstruction and broader continuity results in https://arxiv.org/abs/1102.5306[their original research paper].
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