For a finite abelian group $G$ and a subset $S\subseteq G$ with $0\in S=-S$, let $$ \alpha(S):=\max\{|A|\colon(A-A)\cap S=\varnothing\} $$ and $$ \omega(S):=\max\{|A|\colon A-A\subseteq S\}. $$ These quantities are the independence number and the clique number, respectively, of the Cayley graph induced by $S$ on $G$. Since Cayley graphs are vertex-transitive, we have $$ \alpha(S)\omega(S)\le |G|. \tag{$*$} $$

Let now $$ \alpha^+(S):=\max\{|A|\colon(A+A)\cap S=\varnothing\} $$ and $$ \omega^+(S):=\max\{|A|\colon A+A\subseteq S\}. $$ Is there an analog of the estimate ($*$) for these quantities (maybe, a somewhat weaker one, or with some "restrictions apply" etc.)?

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