Erdős Problem #202 — 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 $1\leq i\leq r$).

The problem asks for the largest number $r$ of pairwise-disjoint residue classes with distinct moduli at most $N$. Erdős and Stein conjectured that the maximum $f(N)$ satisfies $f(N)=o(N)$.

Known results

• Erdős and Szemerédi (1968): established $f(N)=o(N)$ and gave initial upper and lower bounds.
• Croot (2003): improved these to bounds involving $L(N)=\exp(\sqrt{\log N\log\log N})$.
• Chen (2005), then de la Bretèche, Ford, and Vandehey (2013): obtained $N L(N)^{-1+o(1)}<f(N)<N L(N)^{-\sqrt{3}/2+o(1)}$.
• De la Bretèche, Ford, and Vandehey conjectured that $f(N)=N L(N)^{-1+o(1)}$.

Recent claimed proof

The problem history attributes a proof of $f(N)=N L(N)^{-1+o(1)}$ to GPT-5.4 Pro, prompted by Ho Boon Suan, using the Kahn–Kalai conjecture resolution. A recent arXiv paper says a preprint by Ho proves the conjecture, but the retrieved material supplies no independently checkable proof.

Current status (as of July 2026): The classical bounds and the conjectured order are established, while the claimed proof of $f(N)=N L(N)^{-1+o(1)}$ remains unverified.