Problem Statement
For any finite colouring of the integers is there a covering system all of whose moduli are monochromatic?
Categories:
Number Theory Covering Systems
Progress
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.
Source: erdosproblems.com/8 | Last verified: January 13, 2026