# Line free sets correlate locally with dense sets of complexity k-2

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## Introduction

The aim of this page is to generalize the proof that line-free subsets of $[3]^n$ correlate locally with sets of complexity 1 to an analogous statement for line-free subsets of $[k]^n.$

Until this sentence is removed, there is no guarantee that this aim will be achieved, or even that a plausible sketch will result.

## Notation and definitions

The actual result to be proved is more precise than the result given in the title of the page. To explain it we need a certain amount of terminology. Given $j\leq k$ and a sequence $x\in[k]^n,$ let $U_j(x)$ denote the set $\{i\in[n]:x_i=j\}.$ We call this the j-set of x. More generally, if $E\subset[k],$ let $U_E(x)$ denote the sequence $(U_j(x):j\in E).$ We call this the E-sequence of x. For example, if n=10, k=4, $x=3442411123$ and $E=\{2,3\}$ then the E-sequence of x is $(\{4,9\},\{1,10\}).$ It can sometimes be nicer to avoid set-theoretic brackets and instead to say things like that the 24-sequence of x is $(49,235).$

An E-set is a set $\mathcal{A}\subset[k]^n$ such that whether or not x belongs to $\mathcal{A}$ depends only on the E-set of x. In other words, to define an E-set one chooses a collection $\mathcal{U}_E$ of sequences of the form $(U_i:i\in E),$ where the $U_i$ are disjoint subsets of [n], and one takes the set of all x such that $(U_i(x):i\in E)\in\mathcal{U}_E.$ More generally, an $(E_1,\dots,E_r)$-set is an intersection of $E_h$-sets as h runs from 1 to r.

Of particular interest to us will be the sequence $(E_1,\dots,E_{k-1}),$ where $E_j=[k-1]\setminus j.$

To be continued.