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Fix $\epsilon>0$. Every subset $A$ of $\{1,2,\dots,n\}$ with $|A|\geq \epsilon n$ contains a centrally symmetric subset $E$ (in other words, there is a $c$ with $E-c=c-E$) with $|E|> \alpha \epsilon^2 n$. The optimal value for $\alpha$ isn't known, but there are reasonably close upper and lower bounds. Greg Martin's paper, with some coauthor or other, isn't the state of the art, but it's the source that sticks in my mind.