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## co.combinatorics – Visiting zero-sum triples in a vector space

Is it true that for any set $$A\subset\mathbb F_5^n$$ satisfying $$A\cap(-A)=\varnothing$$there is a subset $$A’\subseteq A$$ search any triple $$(a,b,c)\in A\times A\times A$$ with $$a+b+c=0$$ has exactly one or two of its components lying in $$A’$$? That is, no zero-sum triple has all of its components in $$A’$$and no zero-sum triple is completely avoided by $$A’$$.

The same question can be asked for any abelian group, but (the additive group of) $$\mathbb F_5^n$$ is what I presently need. I do not know how relevant the assumption $$A\cap(-A)=\varnothing$$ is.

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