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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 $k=1,D,D^2,\dots$ 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 their original research paper.

If some choice $\omega$ produced a graph with largest component of a graph smaller than $2n/3$, the balanced component cut result with packing parameter one would give an empty graph cut whose sides both have at least $n/3$ vertices. For any fixed such vertex set partition, a uniform edge of the complete graph crosses with probabilityTo avoid crossing in row $i$, at least one of its $k$ offered edges must be noncrossing. The row's probability is $1-q^k\le 1-(4/9{)}^k$. The rows consist of independent random variables, so at $m=cn$ a union bound over at most $2^n$ vertex set partitions givesFor example, the positive integermakes this bound at most $e^{-n}$. Therefore every choice has a component of order at least $2n/3$, with probability tending to one. The simultaneous giant for fixed random-edge choice estimate includes choices made after seeing the entire array; it does not assume a particular online selection rule.

For fixed $0<\lambda <1$ in $G(n,\lambda /n)$, the largest component of a graph has the displayed order with high probability. A dominating Galton-Watson process gives the tail bound $\mathbb{P}(T\ge j)\le e^{-(\lambda -1-\log \lambda )(j-1)}$ by the exponential Markov inequality. For the lower bound, the tree-component expectation in the Erdős-Rényi model diverges at $j=\lfloor (1-\eta )\log n/(\lambda -1-\log \lambda )\rfloor$. The disjoint-set covariance factor and the exclusion of overlaps give relative variance tending to zero, so the second moment method supplies a tree component of that order.