Characterization of order structures avoiding three-term arithmetic progressions

It is known that the set of all nonnegative integers may be equipped with a total order that is chaotic in the sense that there is no monotone three-term arithmetic progressions. Such chaotic order must be so complicated that the resulting ordered set cannot be order isomorphic to the set of all nonnegative integers or the set of all integers with the standard order. In this paper, we completely characterize order structures of chaotic orders on the set of all nonnegative integers, as well as on the set of all integers and on the set of all rational numbers.