Preferential attachment graphs approaching the Rado graph

Let $G'$ be a finite directed graph containing no isolated nodes, and let $f$ be a function satisfying $f(t)\leq t$ for all $t$. Suppose there are constants $0<c_1\leq c_2<1$ such that

$$c_1\cdot t\leq f(t)\leq c_2\cdot t$$

for all sufficiently large $t$. In the graph preferential attachment process ${\textrm{GPA}_f}(G')$, each new vertex attaches according to the degree-proportional rule described in the construction.

Rado graph conjecture. With probability $1$, the infinite limit of ${\textrm{GPA}_f}(G')$ is the Rado graph.

The preceding theorem establishes the analogous statement for directed multigraphs when $f(t)=\Theta(t)$. This conjecture asserts that the result remains true for graphs, where the process selects endpoints without replacement; the restriction $f(t)\leq t$ ensures that this selection is viable.