Problem 687

Erdős Problem #687: declared status 'open'. Formalized: no. Let $Y(x)$ be the maximal $y$ such that there exists a choice of congruence classes $a_p$ for all primes $p\leq x$ such that every integer in $[1,y]$ is congruent to at least one of the $a_p\pmod{p}$. Give good estimates for $Y(x)$. In particular, can one prove that $Y(x)=o(x^2)$ or even $Y(x)\ll x^{1+o(1)}$? Current best: The best known upper bound is due to Iwaniec [Iw78],$$Y(x) \ll x^2.$$The best lower bound is due to Ford, Green, Konyagin, Maynard, and Tao [FGKMT18],$$Y(x) \gg x\frac{\log x\log\log\log x}{\log\log x},$$improving on a previous bound of Rankin [Ra38]. Prize: $1000. OEIS: A048670, A058989. Tags: number theory.