DISPROVED
This has been solved in the negative.
For any finite colouring of the integers is there a covering system all of whose moduli are monochromatic?
Conjectured by Erdős and Graham, who also ask about a density-type version: for example, is\[\sum_{\substack{a\in A\\ a>N}}\frac{1}{a}\gg \log N\]a sufficient condition for $A$ to contain the moduli of a covering system?
The answer (to both colouring and density versions) is no, due to the result of Hough
[Ho15] on the minimum size of a modulus in a covering system - in particular one could colour all integers $<10^{18}$ different colours and all other integers a new colour.
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This page was last edited 05 April 2026. View history
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #8, https://www.erdosproblems.com/8, accessed 2026-04-11
