Anabanti, Chimere (2016) Three questions of Bertram on locally maximal sumfree sets. Technical Report. Birkbeck, University of London, London, UK.

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Abstract
Let G be a finite group, and S a sumfree subset of G. The set S is locally maximal in G if S is not properly contained in any other sumfree set in G. If S is a locally maximal sumfree set in a finite abelian group G, then G = S [ SS [ SS−1 [ pS, where SS = {xy x, y 2 S}, SS−1 = {xy−1 x, y 2 S} and pS = {x 2 G x2 2 S}. Each set S in a finite group of odd order satisfies pS = S. No such result is known for finite abelian groups of even order in general. In view to understanding locally maximal sumfree sets, Bertram asked the following questions: (i) Does S locally maximal sumfree in a finite abelian group imply pS � 2S? (ii) Does there exists a sequence of finite abelian groups G and locally maximal sumfree sets S � G such that SS S ! 1 as G ! 1? (iii) Does there exists a sequence of abelian groups G and locally maximal sumfree sets S � G such that S < cG1 2 as G ! 1, where c is a constant? In this paper, we answer question (i) in the negation, then (ii) and (iii) in affirmation.
Metadata
Item Type:  Monograph (Technical Report) 

Additional Information:  Birkbeck Pure Mathematics Preprint Series #29 
School:  Birkbeck Faculties and Schools > Faculty of Business and Law > Birkbeck Business School 
Depositing User:  Administrator 
Date Deposited:  22 Mar 2019 13:16 
Last Modified:  01 Jul 2024 07:59 
URI:  https://eprints.bbk.ac.uk/id/eprint/26756 
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