Pondering the loose relationship between parametrisation of a hypothesis set and VC dimension

(or the minimum break point,

) led me to the following example and puzzle.

Let the set of points be the natural numbers

Let elements of the hypothesis set be made up of alternating intervals of the same size, like a 1-dimensional checkerboard with varying scale

What is the VC dimension of this hypothesis set?

[there is also a continuous version on the real line, but all the structure is in this simplified version]