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.

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 "$0.9$".

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.

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 "$0.9$".

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.

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.

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 "$0.9$".