Say $f:[n]\times[n]\rightarrow\{-1,1\}$. Here $[n]=\{1,2,\ldots ,n\}$.
Prove that $\mathbb{E}_{x,x',y,y'}f(x,y)f(x,y')f(x',y)f(x',y')\geq\frac1n$.
This is the expectation over all possible values of $x,x',y,y'$. Also $x,x',y,y'$ are iid and uniform.
This expectation is actually a norm as we proved in class, but I don't know how to use that.
I was trying to use the fact that this expectation equals $\mathbb{E}_{x,x'}\left(\left(\mathbb{E}_yf(x,y)f(x',y)\right)^2\right)$.
I was thinking of using the Cauchy-Schwartz inequality somehow. I'm unsure how to continue.