Question:
Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:
i) $|A| = n$, $|B| = m$, and $A$ is a subset of $B$.
ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$.
Determine the minimum value of $m$.
Oh, sometimes I guess this answer is $$m_{\min}=3n-3.$$ This guess relies on this post: How to prove there exists a set $B$ such $\mathrm{card}(B)>\dfrac{n}{3}$? In order to explain this question detail I take an example such that $$A=\{2,4,6,\dots,2n\},B=\{2,4,6,\dots,2n,2n+2,\dots,4n-2\}.$$ This example doesn't solve my problem because $2+(2n+2)=2n+4\in B,2\in A,$ but $2n+2\notin A$.
An upper bound for $m$ is $3n-3$. Let $A=\{k+1,k+2,k+3\dots k+n\}$, $B$ is what is required-$\{k+1,k+2,k+3\dots k+n,2k+3,2k+4,2k+5\dots 2k+2n-1\}$ We need to make $k$ large enough that $4k+7 \gt 2k+2n-1$ or $k \gt n-4$