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The existence (finiteness) of Folkman numbers was established by Folkman (1970) for $r=2$ and by Ne\\v{s}et\\v{r}il and R\\\"odl (1976) for arbitrary $r$, but these proofs led to very weak upper bounds on $f(k;r)$.\n  Recently, Conlon and Gowers and independently the authors obtained a doubly exponential bound on $f(k;2)$. 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