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THE MAXIMUM SIZE OF
$(k,l)$-SUM-FREE SETS IN CYCLIC GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 2 / April 2019
- Published online by Cambridge University Press:
- 26 December 2018, pp. 184-194
- Print publication:
- April 2019
-
- Article
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- You have access
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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].
