Correlation.tex

\section{Line-free sets correlate with insensitive set intersections}

\begin{lemma} Suppose $A \subseteq [k]^n$ satisfies $\eqs{k}(A) \geq \delta$, and let $\gamma > 0$ be a parameter. Then there exists a restriction $(J, x_\barJ)$ with $|J| \geq XXX$ such that one of the following two conditions holds: \begin{enumerate} \item $\eqs{k}^J(A_{x_\barJ}) \geq \delta + \gamma$; \item both $\eqs{k}^J(A_{x_\barJ}) \geq \delta - XXX$ and $\eqs{k-1}^J(A_{x_\barJ}) \geq \delta - XXX$. \end{enumerate} \end{lemma} \begin{proof} Let $r = r(\gamma, n)$ be an integer parameter to be chosen later. As in Lemma~\ref{lem:distributions}, let $J$ be an $[r,4r]$-random subset of $[n]$, let $x$ be drawn from $\eqs{k}^{\barJ}$, let $y$ be drawn from $\eqs{k}^{J}$, and let $z$ be drawn from $\eqs{k-1}^{J}$. We divide into two cases, depending on the value of $\Ex_{J, x}[\eqs{k}(A_{x})^2]$.

\paragraph{Case 1:} $\Ex_{J, x}[\eqs{k}(A_{x})^2] \geq (\delta + \gamma)^2$. In this case, there XXX \\

\paragraph{Case 2:} $\Ex_{J, x}[\eqs{k}(A_{x})^2] < (\delta + \gamma)^2$.

\end{proof}