Problem 202

Erdős Problem #202: declared status 'solved'. Formalized: no. Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one $1\leq i\leq r$). How large can $r$ be in terms of $N$? Current best: Erd\H{o}s believed the lower bound is closer to the truth. These bounds were further improved by Chen [Ch05] and then by de la Bret\'{e}che, Ford, and Vandehey [BFV13] to$$\frac{N}{L(N)^{1+o(1)}}<f(N) < \frac{N}{L(N)^{\sqrt{3}/2+o(1)}}.$$The latter authors conjecture that the lower bound here is the truth. Prize: no. OEIS: A389975. Tags: covering systems.