Erdős Problem #1189 — Extremal questions for irreducible covering sets

Erdős asked for the number of irreducible covering sets of size $k$, extremal values of the largest modulus and reciprocal sum, and infinitely many divisor-set examples. The historical survey records these questions and the example $n=12$.

Known results

• Simpson proved $n_k\leq 2^{k-1}$ for every irreducible covering set of size $k$.
• Sun (2007) showed that the divisors greater than $1$ of $2^{p-1}p$, for every odd prime $p$, form an irreducible covering set; hence infinitely many divisor-set examples exist.
• Balister, Bollobás, Morris, Sahasrabudhe, and Tiba (2024) proved $I(k)\leq\exp\!\left((c+o(1))\frac{k^{3/2}}{(\log k)^{1/2}}\right)$.
• A construction gives $n_k\geq 3\cdot 2^{k-3}$ for $k\geq 5$; Sun’s family gives $n_{2p-1}\geq p2^{p-1}$.

2024 counting result and subsequent construction

The counting question has a substantial upper bound, and the divisor-set question is solved. No complete resolution of the optimal largest modulus or maximal reciprocal sum is reported. GPT-5.4 Thinking is credited for discussion, not for a claimed solution.

Current status (as of March 2026): the divisor-set infinitude question is settled and counting and modulus bounds are known, while the exact count, optimal largest modulus, and maximal reciprocal sum remain open.