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Large sum-free sets in finite vector spaces II

Christian Reiher, Sofia Zotova

Abstract

Answering a question of Leo Versteegen, we prove that for $n\ge 3$ every sum-free set $A\subseteq\mathbb{F}_5^n$ with $|A|\ge 28\cdot 5^{n-3}$ is either contained in the union of two parallel hyperplanes, or isomorphic to $Λ\times \mathbb{F}_5^{n-3}$, where $Λ\subseteq \mathbb{F}_5^3$ denotes a certain sum-free set of size $28$ discovered by Vsevolod Lev and Leo Versteegen.

Large sum-free sets in finite vector spaces II

Abstract

Answering a question of Leo Versteegen, we prove that for every sum-free set with is either contained in the union of two parallel hyperplanes, or isomorphic to , where denotes a certain sum-free set of size discovered by Vsevolod Lev and Leo Versteegen.

Paper Structure

This paper contains 1 section, 1 theorem, 2 equations, 1 figure.

Table of Contents

  1. Introduction

Key Result

Theorem 1.1

If a sum-free set $A\subseteq {\mathds F}_5^2$ with $|A|\ge 5$ is not normal, then it is isomorphic to one of the two sets displayed in Figure fig:5sets. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1.1: Non-normal sum-free sets of size $5$

Theorems & Definitions (2)

  • Theorem 1.1
  • Definition 1.2