Intervals nearly free of integers with a divisor in $(n,2n)$: how large must $y(\epsilon,n)$ be? (Erdős #450)

For $0<\epsilon<1$ and $n\in\mathbb{N}$: how large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$? NOTE: the source (Erdős–Graham) does not specify the quantifier on $x$, and pinning down the intended reading is part of the problem — the natural candidates are 'for all $x$' and 'for some/typical $x$'; the known partial results below show the answer differs drastically between readings and $\epsilon$-regimes.

Acceptance. FULLY RESOLVES: under an explicitly stated quantifier reading (for-all-$x$, or for-some-$x$/all-sufficiently-large-$x$ — address at least the for-all reading, and state which is used), determine the asymptotic order of $y(\epsilon,n)$ for each regime of $\epsilon=\epsilon(n)$, including a proof of the exact threshold in $\epsilon$ below/above which no finite $y$ exists; machine-checkable (Lean/Coq) preferred, else a complete written proof. ADVANCES: (a) sharpen the nonexistence criterion — replace the sufficient condition stated in the background by a weaker one, or prove a matching existence result, with proof; (b) determine $y(\epsilon,n)$ up to constants in any part of the intermediate regime $1/n\ll\epsilon\ll(\log n)^{-\delta}(\log\log n)^{-3/2}$, with proof; (c) prove the analogous threshold behaviour under the exists-$x$ reading, connecting to the maximal-gap problem; (d) reproducible computations for a range of small $n$ that exactly determine, for each window length $y$, the extremal counts $\max_x$ and $\min_x$ of integers in $(x,x+y)$ with a divisor in $(n,2n)$ (with code and exhaustiveness certificate over the relevant periodic range), yielding certified small-$n$ values of $y(\epsilon,n)$ under both readings. Deliver the proof file, or the code + certified tables.