Characterization of order structures avoiding three-term arithmetic progressions

TL;DR

This paper characterizes complex order structures on nonnegative integers, integers, and rationals that avoid three-term arithmetic progressions, revealing the nature of chaotic orders in these sets.

Contribution

It provides a complete characterization of chaotic order structures avoiding three-term arithmetic progressions on key countable sets.

Findings

Chaotic orders cannot be isomorphic to standard integer or nonnegative integer orders.

Complete classification of such orders on nonnegative integers, integers, and rationals.

Insights into the structure of orderings avoiding three-term arithmetic progressions.

Abstract

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.