Maximal subsets free of arithmetic progressions in arbitrary sets

TL;DR

This paper investigates the largest subset free of arithmetic progressions within any set of size n, showing that focusing on the initial interval [1, n] suffices, building on prior foundational work.

Contribution

It demonstrates that to find maximal progression-free subsets, it is enough to consider the initial segment [1, n], extending previous research.

Findings

Maximal progression-free subsets can be found within the interval [1, n].

The result simplifies the search for large progression-free subsets.

Builds upon and extends the work of Komlós, Sulyok, and Szemerédi.

Abstract

We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval $[1,\dots, n]$. The study continues the work of Koml\'os, Sulyok, and Szemer\'edi.