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THE MAXIMUM SIZE OF $(k,l)$-SUM-FREE SETS IN CYCLIC GROUPS

Part of: Extremal combinatorics Abelian groups Sequences and sets Additive number theory; partitions

Published online by Cambridge University Press:  26 December 2018

BÉLA BAJNOK Open the ORCID record for BÉLA BAJNOK [Opens in a new window]  and
RYAN MATZKE Open the ORCID record for RYAN MATZKE [Opens in a new window]
BÉLA BAJNOK*
Affiliation:
Department of Mathematics, Gettysburg College, Gettysburg, PA, USA email bbajnok@gettysburg.edu
RYAN MATZKE
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN, USA email matzk053@umn.edu
*
✉bbajnok@gettysburg.edu
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Abstract

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A subset $A$ of a finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ never equals the sum of $l$ (not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a $(k,l)$-sum-free subset in $G$ for all $k$ and $l$ in the case when $G$ is cyclic by proving that it suffices to consider $(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’, Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a $(k,l)$-sum-free subset of an abelian group’, Int. J. Number Theory 5(6) (2009), 953–971].

Keywords

sum-free setsabelian groupsKneser’s theoremarithmetic progressions

MSC classification

Primary: 11B75: Other combinatorial number theory
Secondary: 05D99: None of the above, but in this section 11B25: Arithmetic progressions 11P70: Inverse problems of additive number theory, including sumsets 20K01: Finite abelian groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 2 , April 2019 , pp. 184 - 194
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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