Estimate $F_k(p_1,\ldots,p_u)$: multiples of some $p_i$ forced in every length-$k$ interval (Erdős #1143)

Let $p_1<p_2<\cdots<p_u$ be primes and let $k\geq 1$. Define $F_k(p_1,\ldots,p_u)$ to be the largest integer $F$ such that every interval of $k$ consecutive positive integers contains at least $F$ integers divisible by at least one of the $p_i$ — equivalently, $F_k$ is the minimum, over all length-$k$ intervals, of the number of elements of the interval divisible by some $p_i$. Estimate $F_k(p_1,\ldots,p_u)$, particularly in the range $k=\alpha p_u$ for a constant $\alpha>2$.

Acceptance. FULLY RESOLVES: a proof giving the correct order (ideally the exact value or asymptotic) of $F_k(p_1,\ldots,p_u)$ throughout the range $k=\alpha p_u$ with $\alpha>2$ — in particular a rigorous determination for $\alpha>3$, where the background records essentially nothing known; machine-checkable proof preferred, else full written proof. ADVANCES: rigorously establish or reconstruct, with a complete proof, the exact bound for $2<\alpha<3$ (given that the claimed Erdős–Selfridge result is unlocated); prove new upper or lower bounds on $F_k$ for some regime $\alpha>3$; or compute $F_k$ exactly for a family of prime sets by exploiting the period-$\prod_i p_i$ structure and certify the values, yielding a data-backed conjecture with a reproducible program. Deliver the proof, or the exact-computation code plus the certified $F_k$ values and the parameter range covered.