It is possible that a more technical version of this question has been asked and answered in the literature. If so, then a reference is much appreciated. I will phrase it in terms of colored tapes placed on a number line.
I have an unlimited supply of transparent but partially colored arithmetic progression tapes, one for each prime number. Thus, the tape for 3 looks a lot like the tape for 29, except instead of having a blue square at every third inch, I have a yellow square at every 29th inch; the tape is clear otherwise. I am going to make several arrangements of tape. My first arrangement produces a color coding of integers similar to indicating divisibility, except I perversely decide to line up all the colors at 1. This leaves positions 0 and 2 with no colors, and every other position gets finitely many colors (88 gets only blue and yellow).
What other arrangements can I make? I can pick any nonempty set of integers and arrange that each get some of a partition of colors, although the integer may get more colors than I assigned. I can sometimes arrange to leave a square uncolored, regardless of how the primes are partitioned. That is not the main question though.
Main question: can I arrange that every integer gets only finitely many colors?
An amusing fact (courtesy of the Chinese Remainder Theorem) is that, with a finite number of such tapes and modulo translation, every such arrangement looks the same. Thus I do not expect a finite upper bound to the number of colors received by an integer.
Although I would appreciate a specific example, I am also interested in knowing how weak a subtheory of ZF or PA is needed to prove existence of such an arrangement. (I harbor a suspicion that PRA will resolve it negatively if it resolves at all.) Specific references to the literature on covering congruences are welcome (as are alternate search terms), but for this problem I am considering prime moduli only.
Gerhard "Your Responses Are Being Recorded" Paseman, 2014.03.14