The problem described isn't a metaphor or a proxy; it's not about some other thing like mutual information. It's about exactly what it says it is: evenly-space

The problem described isn't a metaphor or a proxy; it's not about some other thing like mutual information. It's about exactly what it says it is: evenly-spaced (arithmetic) subsequences. More formally, given k, you have the set {1, ..., N}, and you want to produce a map from {1, ..., N} to a two-element set (which we can denote {blue, red}) such that there are no blue arithmetic sequences of length at least 3, and there are no red arithmetic sequences of length at least k. The question then becomes, given k, how large can N be with this remaining possible?