, Volume 78, Issue 1-2, pp 101-112

Extremal problems on sum-free sets and coverings in tridimensional spaces

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Given a prime power q, define c(q) as the minimum cardinality of a subset H of \({\mathbb{F}}^3_q\) which satisfies the following property: every vector in this space differs in at most 1 coordinate from a multiple of a vector in H. In this work, we introduce two extremal problems in combinatorial number theory aiming to discuss a known connection between the corresponding coverings and sum-free sets. Also, we provide several bounds on these maps which yield new classes of coverings, improving the previous upper bound on c(q).

Manuscript received: March 12, 2008 and, in final form, June 17, 2009.