Partition of a Set Which Contains an Infinite Arithmetic (Respectively Geometric) Progression

TL;DR

This paper proves that in any two-way partition of a set containing an infinite arithmetic or geometric progression, at least one subset contains infinitely many triplets forming such progressions.

Contribution

It establishes a new combinatorial result about the distribution of progressions within partitions of infinite sets.

Findings

At least one subset contains infinitely many arithmetic triplets.

At least one subset contains infinitely many geometric triplets.

The result applies to any partition of sets with infinite progressions.

Abstract

We prove that for any partition of a set which contains an infinite arithmetic (respectively geometric) progression into two disjoint subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression.