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

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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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