= Bounded exceptional primes for least quadratic nonresidues
If, for every $\eta>0$, the <smooth numbers> up to $T$ with all <prime factors> at most $T^\eta$ have density at least $\kappa(\eta)>0$, then for fixed $\varepsilon>0$ there are only $O_\varepsilon(1)$ <primes> $p\leq N$ with $n(p)>N^\varepsilon$. Apply the <variance form of the large sieve> to the <indicator function> of the $N^\varepsilon$-smooth numbers up to $N^2$. For every exceptional <prime>, all these numbers are nonzero <quadratic residues>, so half its nonzero <residue classes> are empty. Their variance contribution is at least one third of the square of the total count. The large sieve bounds the sum of contributions by a constant times $N^2$ times that count. The density assumption with $\eta=\varepsilon/2$ finishes the argument.
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