Unbounded logarithmic limsup in Erdős problem 684

ABSTRACT For \(0\le k\le n\), write \(\binom{n}{k}=uv\) 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 \(n_{M}=tL_{M}-1\), where \(L_{M}=\operatorname{lcm}(1,\ldots,M)\) 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
\[
  \limsup_{n\to\infty}\frac{f(n)}{\log n}=\infty.
\]
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-\(\Omega\) arithmetic-progression estimate, including a \(Q_{M}=M!/L_{M}\) 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.