Erdős Problem #1202

Let \(\epsilon > 0\). Does there exist \(k\) such that, for every \(n\), any primes \(p_1 < \cdots < p_k < n^{1-\epsilon}\) and any sets \(A_i\) of \(\frac{p_i - 1}{2}\) residue classes modulo \(p_i\), the number of \(m \le n\) that avoid \(A_i \pmod{p_i}\) for every \(i\) is at most \(\epsilon n\)? The answer is negative: according to [erdosproblems.com/1202](https://www.erdosproblems.com/1202), Price (with GPT-5.4 Pro) gave a construction in which the \(A_i\) are intervals of residues and at least \((1/2 - c) n\) integers in \([1, n]\) survive. The large sieve gives a positive answer when \(p_k < n^{1/2}\) instead. See also `Green44` for a related special case.