Unbounded logarithmic limsup in Erdős problem 684

For $0\le k\le n$, write $\left(\genfrac{}{}{0px}{}{n}{k}\right)=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erdős problem 684 asks for bounds on $f(n)$. We resolve the problem at the order level. By a short-multiplier construction $n_M=t{L}_M-1$, where $L_M=\mathrm{lcm}(1,\dots ,M)$ and $t$ is a multiplier of size $\exp (o(M))$ extracted from a Fourier sieve, we prove that for every fixed $C>1$ there exist integers $n$ with $$f(n)>(C-o(1))\log n,$$ hence $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{f(n)}{\log n}=\mathrm{\infty }\mathrm{.}$$ We thus refute the widely expected upper bound $f(n)\ll \log n$ and place the order of $f(n)$ strictly above $\log n$ infinitely often. A matching polylogarithmic upper bound $f(n)\ll (\log n{)}^2$ is known by Alexeev, Putterman, Sawhney, Sellke, and Valiant (arXiv:2603.29961). The reduction of the multiplier sieve to a dyadic fixed-$\mathrm{\Omega }$ arithmetic-progression estimate, including a $Q_M=M!\mathrm{/}{L}_M$ box parametrization, a local harmonic-height cap, and an exact-$a$ product-shell extraction, is new. The required estimate uses Timofeev’s mean-in-progressions framework together with a Burgess-based mod-$p$ saving on the relevant prime band.