Complete 3-term arithmetic progression free sets of small size in vector spaces and other abelian groups

2024/01/11 by Bence Csajbók, Zoltán Lóránt Nagy, Csajbók, Bence +1
Mathematics · #05D05 #11B25 #51E21 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2401.06283

openalex publication_date 2024/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A subset S of an abelian group G is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, S is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.

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