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][1], with some coauthor or other, isn't the state of the art, but it's the source that sticks in my mind.

Personally, I prefer the continuous statement: every subset of $[0,1]$ with measure $\epsilon$ contains a centrally symmetric subset with measure $0.6\epsilon^2$. And the the "$0.6$", while not best possible, cannot be replaced with "$0.9$".


  [1]: http://front.math.ucdavis.edu/0807.5121